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This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 101. |
निम्न समाकलों के मान ज्ञात कीजिए - (i) `int_(0)^(pi//2)cos^(3)x sin x dx` |
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Answer» Correct Answer - `(1)/(4)` माना `cos x= t rArr - sin x dx =dt` |
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| 102. |
`int_(pi//4)^(pi//2)e^(x)(logsinx+cotx)dx` का मान हैA. `e^(pi//4)log2`B. `-e^(pi//4)log2`C. `(1)/(2)e^(pi//4)log2`D. `-(1)/(2)e^(pi//4)log2` |
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Answer» Correct Answer - D `I=int_(pi//4)^(pi//2)e^(x)(logsinx+cotx)dx` `=[e^(x)logsinx]_(pi//4)^(pi//2)-int_(pi//4)^(pi//2)cotx dx+int_(pi//4)^(pi//2)e^(x)cotxdx` `=[e^(x)logsinx]_(pi//4)^(pi//2)` `=[e^(pi//2)log1-e^(pi//4)logsqrt(2)]` `=-(1)/(2)e^(pi//4)log2` |
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| 103. |
निम्न समाकलों के मान ज्ञात कीजिए - (i) `int_(0)^(pi//4)sqrt(tan theta) d theta` |
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Answer» Correct Answer - `(1)/(2sqrt2)[log[((sqrt2-1))/((sqrt2+1))]+pi]` माना `tan theta= t^(2) rArr sec^(2) theta d theta = 2 t dt` तब ` int_(0)^(pi//4)sqrt(tan theta) d theta = int_(0)^(1)(2t^(2))/(1+t^(4))dt=int_(0)^(1)((t^(2)+1)(t^(2)-1))/(t^(4)+1)dt` `=int_(0)^(1)((1+(1)/(t^(2))))/((t^(2)+(1)/(t^(2))))dt-int_(0)^(1)((1-(1)/(t^(2))))/((t^(2)+(1)/(t^(2))))dt` `=int_(0)^(1)((1+(1)/(t^(2))))/((t-(1)/(t))^(2)+(sqrt2)^(2))dt-int_(0)^(1)((1-(1)/(t^(2))))/((t+(1)/(t))^(2)-(sqrt2)^(2))dt` अब माना `t-(1)/(t)=u` प्रथम समाकलन में तथा `t+(1)/(t)=v` द्वितीय समाकलन में |
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| 104. |
समाकलन `int_(0)^(1)(1-x)^(9)dx` का मान क्या है ?A. `(100)/(110)`B. `(10)/(111)`C. `(1)/(110)`D. `-(110)/(100)` |
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Answer» माना `I=int_(0)^(1)x(1-x)^(9)dx[becauseint_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx]` `rArrI=int_(0)^(1)(1+0-x)(1-1-0+x)^(9)dx` `rArrI=int_(0)^(1)(1-x)x^(9)dx=int_(0)^(1)(x^(9)-x^(10))dx` `rArrI=[(x^(10))/(10)-(x^(11))/(11)]_(0)^(1)rArrI=[(1)/(10)-(1)/(11)]rArrI=(1)/(110)` 6. `int_(0)^(2a)f(x)dx={{:(2int_(0)^(a)f(x)dx",","यदि"f(2a-x)=f(x)),(0",","यदि"f(2a-x)=-f(x)):}` 7. `int_(-a)^(a)f(x)dx={{:(2int_(0)^(a)f(x)dx",","यदि"f(-x)=f(x)"सम फलन"),(0",","यदि"f(-x)=-f(x)"विषम फलन"):}` |
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| 105. |
निम्न समाकलों के मान ज्ञात कीजिए - (i) `int_(0)^(a)(x)/(sqrt(a^(2)+x^(2)))dx` |
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Answer» Correct Answer - `a(sqrt2-1)` माना `x^(2)+a^(2)=t rArr 2xdx=dt` तब `I=int_(0)^(a)(x)/(sqrt(a^(2)+x^(2)))dx=int_(a^(2))^(2a^(2))(dt)/(2sqrtt)` `=(1)/(2)[(sqrtt)/(1//2)]_(a^(2))^(2a^(2))=a(sqrt2-1)` |
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| 106. |
निम्न का समाकलन कीजिए - `int_(1)^(2)(logx)/(x^(2))dx` |
| Answer» Correct Answer - `(1-log2)/(2)` | |
| 107. |
`=int_(-pi//2)^(pi//2)sin^(2)xdx` का मान हैA. `(pi)/(2)`B. `pi`C. `(pi)/(2)+(1)/(2)`D. `pi+1` |
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Answer» Correct Answer - A माना `f(x)=sin^(2)x` `f(-x)=sin^(2)(-x)=sin^(2)x=f(x)` `thereforeint_(-pi//2)^(pi//2)sin^(2)xdx=2int_(0)^(pi//2)sin^(2)xdx` `(2lceiling((2+1)/(2))lceiling((1)/(2)))/(2lceiling((2+0+2)/(2)))` `=(2xx(1)/(2)sqrt(pi)xxsqrt(pi))/(2)=(pi)/(2)` |
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| 108. |
निम्न समाकलों के मान ज्ञात कीजिए - (i) `int_(-pi//4)^(pi//4)"cosec"^(2)xdx` |
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Answer» Correct Answer - `-2` `I=int_(-pi//4)^(pi//4)"cosec"^(2)xdx=2int_(0)^(pi//4)"cosec"^(2)xdx` `=2[-cot x]_(0)^(pi//4)=-2[cot.(pi)/(4)-cot 0]=-2` |
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| 109. |
निम्न का समाकलन कीजिए - `int_(0)^(1)(dx)/(1+x+x^(2))` |
| Answer» Correct Answer - `(pi)/(3sqrt3)` | |
| 110. |
समाकलन `int_(-1)^(1)|x|dx` का मान क्या है ?A. 1B. 0C. 2D. `-1` |
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Answer» Correct Answer - A `becausef|x|` यहाँ एक सम फलन है अर्थात `f(-x)=|-x|=|x|=f(x)` `thereforeint_(-1)^(1)|x|dx=2int_(0)^(1)|x|dx=2int_(0)^(1)xdx` `=[(x^(2))/(2)]_(0)^(1)=2*(1)/(2)[x^(2)]_(0)^(1)=(1)^(2)-(0)^(2)=1` |
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| 111. |
निम्न का समाकलन कीजिए - `int_(2)^(0)sqrt(6x+4)dx` |
| Answer» Correct Answer - `(56)/(9)` | |
| 112. |
निम्न का समाकलन कीजिए - `int_(0)^(pi//2)(sin^(7)x)/(sin^(7)x+cos^(7)x)dx` |
| Answer» Correct Answer - `(pi)/(4)` | |
| 113. |
निम्न का समाकलन कीजिए - `int_(0)^(pi//2)(cos^(3)x)/(sin^(3)x+cos^(3)x)dx` |
| Answer» Correct Answer - `(pi)/(4)` | |
| 114. |
निम्न का समाकलन कीजिए - `int_(0)^(pi//2)cos^(3)xdx` |
| Answer» Correct Answer - `(2)/(3)` | |
| 115. |
निम्न का समाकलन कीजिए - `int_(0)^(pi//4)sin 2x sin 3xdx` |
| Answer» Correct Answer - `(3sqrt2)/(10)` | |
| 116. |
`int_(0)^(2pi)sin^(5)((x)/(4))dx` किसके बराबर है ?A. `8//15`B. `16//15`C. `32//15`D. 0 |
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Answer» Correct Answer - C दिया है ,`int_(0)^(2pi)sin^(5)((x)/(4))dx` `(x)/(4)=trArrdx=4dt` रखने पर, `=int_(0)^(pi/2)sin^(5)t*4dt=4int_(0)^(pi/2)sin^(5)t dt` `=(4*4*2)/(5*3)=(32)/(15)` [वॉली सूत्र से] |
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| 117. |
निम्न का समाकलन कीजिए - `int_(0)^(pi//2)sqrt(1+cos 2x)dx` |
| Answer» Correct Answer - `sqrt2` | |
| 118. |
`int_(0)^(pi//2)e^(x)sinxdx` का मान हैA. `(1)/(2)(e^(pi//2)-1)`B. `(1)/(2)(e^(pi//2)+1)`C. `(1)/(2)(1-e^(pi//2))`D. `2(e^(pi//2)+1)` |
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Answer» Correct Answer - B `I=int_(0)^(pi//2)e^(x)sinxdx` `=[(e^(x))/(2)(sinx-cosx)]_(0)^(pi//2)` `[becauseinte^(ax)sinbxdx=(e^(ax))/(a^(2)+b^(2))(asinbx-bcosbx)]` `=[(e^(pi//2))/(2)(1-0)-(1)/(2)(0-1)]=(1)/(2)(e^(pi//2)+1)` |
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| 119. |
निम्न का समाकलन कीजिए - `int_(0)^(a)sin^(-1)sqrt((x)/(a+x))dx` |
| Answer» Correct Answer - `a((pi)/(2)-1)` | |
| 120. |
समाकलनों`I_(1)=int_(pi//6)^(pi//3)(dx)/(1+sqrt(tanx))` और `I_(2)=int_(pi//6)^(pi//3)(sqrt(sinx)dx)/(sqrt(sinx)+sqrt(cosx))` पर विचार कीजिए। `I_(1)-I_(2)` किसके तुल्य है ? |
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Answer» Correct Answer - A दिया है, `I_(1)=int_(pi//6)^(pi//3)(dx)/(1+sqrt(tanx))` तथा `I_(2)=int_(pi//6)^(pi//3)(sqrt(sinx))/(sqrt(sinx)+sqrt(cosx))dx` . . . (i) `I_(1)` को निम्न प्रकार लिखा जा सकता है `I_(1)=int_(pi//6)^(pi//3)(dx)/(1+(sqrt(sinx))/(sqrt(cosx)))` `=int_(pi//6)^(pi//3)(sqrt(cosx))/(sqrt(sinx)+sqrt(cosx))dx`. . . (ii) अब, `I_(1)-I_(2)=int_(pi//6)^(pi//3)(sqrt(cosx))/(sqrt(sinx)+sqrt(cosx))dx` `-int_(pi//6)^(pi//3)(sqrt(sinx))/(sqrt(sinx)+sqrt(cosx))dx` `=int_(pi//6)^(pi//3)((sqrt(cosx)-sqrt(sinx))/(sqrt(cosx)+sqrt(sinx)))dx` `=int_(pi//6)^(pi//3)` `(sqrtcos((pi)/(3)+(pi)/(6)-x)-sqrt(sin((pi)/(3)+(pi)/(6)-x)))/(sqrt(cos((pi)/(3)+(pi)/(6)-x))+sqrt(sin((pi)/(3)+(pi)/(6)-x)))dx` `[becauseint_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx]` `=int_(pi//6)^(pi//3)(sqrt(cos((pi)/(2)-x))-sqrt(sin((pi)/(2)-x)))/(sqrt(cos((pi)/(2)-x))+sqrt(sin((pi)/(2)x)))dx` `=int_(pi//6)^(pi//3)(sqrt(sinx)-sqrt(cosx))/(sqrt(sinx)+sqrt(cosx))dx` `=-{int_(pi//6)^(pi//3)(sqrt(cosx)-sqrt(sinx))/(sqrt(sinx)+sqrt(cosx))dx}` `=-(I_(1)-I_(2)rArr2(I_(1)-I_(2))=0` `rArrI_(1)-I_(2)=0` |
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| 121. |
समाकलनों `I_(1)=int_(pi//6)^(pi//3)(dx)/(1+sqrt(tanx))` और `I_(2)=int_(pi//6)^(pi//3)(sqrt(sinx)dx)/(sqrt(sinx)+sqrt(cosx))` पर विचार कीजिए । `I_(1)` किसके तुल्य है ?A. `pi//24`B. `pi//18`C. `pi//12`D. `pi//6` |
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Answer» Correct Answer - C दिया है, `I_(1)=int_(pi//6)^(pi//3)(dx)/(1+sqrt(tanx))` तथा `I_(2)=int_(pi//6)^(pi//3)(sqrt(sinx))/(sqrt(sinx)+sqrt(cosx))dx` . . . (i) `I_(1)` को निम्न प्रकार लिखा जा सकता है `I_(1)=int_(pi//6)^(pi//3)(dx)/(1+(sqrt(sinx))/(sqrt(cosx)))` `=int_(pi//6)^(pi//3)(sqrt(cosx))/(sqrt(sinx)+sqrt(cosx))dx`. . . (ii) समी (i) से , `I_(1)=int_(pi//6)^(pi//3)(sqrt(cosx))/(sqrt(sinx)+sqrt(cosx))dx` . . . (iii) `=int_(pi//6)^(pi//3)` `(sqrt(cos((pi)/(3)+(pi)/(6)-x)))/(sqrt(sin((pi)/(3)+(pi)/(6)-x))+sqrt(cos((pi)/(3)+(pi)/(6)-x)))dx` `=int_(pi//6)^(pi//3)(sqrt(cos((pi)/(2)-x)))/(sqrt(sin((pi)/(2)x))+sqrt(cos((pi)/(2)-x)))dx` `=int_(pi//6)^(pi//3)(sqrt(sinx))/(sqrt(cosx)+sqrt(sinx))dx` . . . (iv) समी (iii) तथा (iv) को जोड़ने पर, `2I_(1)=int_(pi//6)^(pi//3)((sqrt(cosx)+sqrt(sinx))/(sqrt(cosx)+sqrt(sinx)))dx` `2I_(1)=int_(pi//6)^(pi//3)(1)dx=[x]_(pi//6)^(pi//3)` `=((pi)/(3)-(pi)/(6))=(pi)/(6)` `rArrI_(1)=(pi)/(12)` |
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| 122. |
यदि `0ltaltb`, तो `int_(a)^(b)(|x|)/(x)dx` किसके बराबर है ?A. `|b|-|a|`B. `|a|-|b|`C. `|b|//|a|`D. 0 |
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Answer» Correct Answer - A दिया है, `int_(a)^(b)(|x|)/(x)dx=int_(a)^(b)dx[because0ltaltb]` `=[x]_(a)^(b)=|b|-|a|` |
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| 123. |
समाकलनों `A=int_(0)^(pi)(sinxdx)/(sinx+cosx)` और `B=int_(0)^(pi)(sinxdx)/(sinx-cosx)` पर विचार कीजिए। B का मान क्या है ?A. `pi//4`B. `pi//2`C. `3pi//4`D. `pi` |
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Answer» Correct Answer - B माना `I=A=int_(0)^(pi)(sinx)/(sinx+cosx)dx`. . . (i) तथा `I=int_(0)^(pi)(sinx)/(sinx-cosx)dx` . . . (ii) `[becauseint_(0)^(a)f(x)dx=int_(0)^(pi)f(a-x)dx]` समी (i) व (ii) को जोड़ने पर, `2I=int_(0)^(pi)((sinx)/(sinx+cosx)+(sinx)/(sinx-cosx))dx` `rArr2I=` `int_(0)^(pi)(sinx(sinx-cosx+sinx+cosx))/(sin^(2)x-cos^(2)x)` `rArr2I=int_(0)^(pi)(2sin^(2)x)/(sin^(2)x-cos^(2)x)dx` `rArr2I=4int_(0)^(pi//2)(sin^(2)x)/(sin^(2)x-cos^(2)x)dx` . . . (iii) `[becauseint_(0)^(2a)f(x)dx=2int_(0)^(a)f(x)dx]` `rArr2I=4int_(0)^(pi//2)(cos^(2)x)/(cos^(2)x-sin^(2)x)dx` . . . (iv) समी (iii) व (iv) को जोड़ने पर, `4I=4int_(0)^(pi//2)((sin^(2)x-cos^(2)x)/(sin^(2)x-cos^(2)x))dx` `rArr4I=4[x]_(0)^(pi//2)rArr4I=4xx(pi)/(2)rArrI=(pi)/(2)` |
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| 124. |
`int_(2)^(4){|x-2|+|x-3|}dx` का मान है -A. 1B. 2C. 3D. 5 |
| Answer» Correct Answer - C | |
| 125. |
समाकलनों `A=int_(0)^(pi)(sinxdx)/(sinx+cosx)` और `B=int_(0)^(pi)(sinxdx)/(sinx-cosx)` पर विचार कीजिए। निम्नलिखित में कौन -सा एक सही है ?A. A=2BB. B=2AC. A=BD. A=3B |
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Answer» Correct Answer - C दिया है, `A=int_(0)^(pi)(sinx)/(sinx+cosx)dx` और `B=int_(0)^(pi)(sinx)/(sinx-cosx)dx` अब, `A=int_(0)^(pi)(sin(pi-x))/(sin(pi-x)+cos(pi-x))dx` `[becauseint_(0)^(a)f(x)dx=int_(0)^(a)f(a-x)dx]` `=int_(0)^(pi)(sinxdx)/(sinx-cosx)=B` |
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| 126. |
निम्न का समाकलन कीजिए - `int_(0)^(a)(sqrtx)/(sqrtx+sqrt(a-x))dx` |
| Answer» Correct Answer - `(pi)/(4)` | |
| 127. |
यदि `b lt a`, तब `int_(a)^(b)(dx)/(sqrt((x-a)(b-x)))` का मान है -A. `(pi)/(2)`B. `pi`C. `(pi)/(2)(b-a)`D. `(pi)/(4)(b-a)` |
| Answer» Correct Answer - B | |
| 128. |
मान लीजिए `I=int_(0)^(1)(sinx)/(sqrtx)dx` तथा `J=int_(0)^(1)(cosx)/(sqrtx)dx`, तब निम्न में से कौन-सा कथन सही है?A. `I gt(2)/(3)" व "J lt 2`B. `I gt (2)/(3)" व "J gt2`C. `I gt (2)/(3)" व " J lt 2`D. `I lt (2)/(3)" व " J gt 2` |
| Answer» Correct Answer - C | |
| 129. |
निम्न समाकलों के मान ज्ञात कीजिए - (ii) `int_(0)^(pi//2)(1)/(1+cot x)dx` |
| Answer» Correct Answer - `(pi)/(4)` | |
| 130. |
`int_(0)^(2)(x^(3)dx)/((x^(2)+1)^(3//2))` का मान है -A. `(sqrt2-1)^(2)`B. `((sqrt2-1)^(2))/(sqrt2)`C. `(sqrt2-1)/(sqrt2)`D. इनमे से कोई नहीं |
| Answer» Correct Answer - D | |
| 131. |
निम्न समाकलों के मान ज्ञात कीजिए - (ii) `int_(1)^(2)(d)/(xsqrt(x^(2)-1))` |
| Answer» Correct Answer - `(pi)/(3)` | |
| 132. |
निम्न समाकलों के मान ज्ञात कीजिए - (iv) `int_(1)^(2)(dx)/(x(1+x^(2)))` |
| Answer» Correct Answer - `(3)/(2)log 2-(1)/(2)log5` | |
| 133. |
निम्न समाकलों के मान ज्ञात कीजिए - (v) `int_(1)^(e)(e^(x))/(x)(1_x log x)dx` |
| Answer» Correct Answer - `e^(e)` | |
| 134. |
निम्न समाकलों के मान ज्ञात कीजिए - (iii) `int_(pi//4)^(pi//2)e^(x)(log sin x+cot x)dx` |
| Answer» Correct Answer - `e^(pi//4)logsqrt2` | |
| 135. |
`int_(0)^(a)(x^(4)dx)/((a^(2)+x^(2))^(4))` का मान है -A. `(1)/(16a^(3))((pi)/(4)-(1)/(3))`B. `(1)/(16a^(3))((pi)/(4)+(1)/(3))`C. `(1)/(16)a^(3)((pi)/(4)-(1)/(3))`D. `(1)/(16)a^(3)((pi)/(4)+(1)/(3))` |
| Answer» Correct Answer - A | |
| 136. |
निम्नलिखित कथनों पर विचार कीजिए I. `lim_(n-oo)[(1)/(n+1)+(1)/(n+2)+...+(1)/(1+2n)]` II. `lim_(n-infty)((1)/(2n+1)+(1)/(2n+2)+...(1)/(6n))` का मान `log_(3)` है। उपरोक्त कथनों में से कौन -सा /से कथन सत्य है /हैं ?A. केवल IB. केवल IIC. I और II दोनोंD. न तो I और न ही II |
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Answer» I. `I=underset(n-infty)(lim)(1)/(n)[(1)/(1+(1)/(n))+(1)/(1+(2)/(n))+...+(1)/(1+(n)/(n))]` `=underset(n-infty)(lim)(1)/(n)sum_(r=1)^(n)(1)/((1+(r)/(n)))` मानक रूप मान लीजिए `(r)/(n)=x` तथा `(1)/(n)=dx` जब,`r=1,x=(1)/(n)to0` जब, `r=n,x=(n)/(n)to` 1 जबकि `ntoinfty` `thereforeI=int_(0)^(1)(1)/(1+x)dx=[log(1+x)]_(0)^(1)=log2-log1=log2` II. `S_(n)=((1)/(2n+1)+(1)/(2n+2)+...+(1)/(6n))` `=sum_(r=1)^(4n)(1)/(2n+r)=sum_(r=1)^(4n)*(1)/(n)*(1)/(2+((r)/(n)))` `rArrS=underset(ntoinfty)(lim)S_(n)=int_(0)^(4)(dx)/(2+x)=[log|2+x|]_(0)^(4)=log6-log2` `thereforeS=log3` |
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| 137. |
निम्न समाकलों के मान ज्ञात कीजिए - (iv) `int_(0)^(oo)log(x+(1)/(x))(dx)/(1+x^(2))` |
| Answer» Correct Answer - `pilog_(e)2` | |
| 138. |
`int_(0)^(1000)e^(x-[x])dx` का मान है -A. `e^(1000)-1`B. `(e^(1000)-1)/(e-1)`C. `1000(e-1)`D. `(e-1)/(1000)` |
| Answer» Correct Answer - C | |
| 139. |
`lim_(n to oo)sum_(r=1)^(n)e^(r//n)` का मान है -A. `e`B. `e-1`C. `1-e`D. `e+1` |
| Answer» Correct Answer - B | |
| 140. |
`lim_(x to 0)(1)/(x^(3))int_(0)^(x)(log(1+t))/(t^(4)+4)dt` का मान निम्न है- |
| Answer» Correct Answer - B | |
| 141. |
इन समाकलनो का मान कीजिएः `int_(0)^(pi//4) tan^2xdx` |
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Answer» `underset(0)overset(pi//4)int tanx^(2)xdx` `underset(0)overset(pi//4)int (sec^(2)x-1)dx=[tan x-x]_(0)^(pi//4)` `=[tan ""(pi)/(4)-pi/4]-[tan 0-0]` `=1-pi/4` |
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| 142. |
इन समाकलनो का मान कीजिएः `int_(0)^(pi//4) sin3x sin 2xdx` |
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Answer» `underset(0)overset(pi//4)int sin3x sin 2dx` `=1/2underset(0)overset(pi//4)int (2 sin 3 x sin2x)dx` `=1/2underset(0)overset(pi//4)int (cos x-cos 5x)dx` `[therefore 2 sin A sin B=cos (A-B)-cos(A+B)]` `=1/2[sin x-(sin 5x)/(5)]_(0)^(pi//4)` `=1/2[1/sqrt2+1/5sin""(pi)/(4)]=1/2[1/sqrt2+1/(2sqrt2)]` `=(6)/(2(5sqrt2))=(3sqrt2)/(10)` |
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| 143. |
इन समाकलनो का मान कीजिएः `int_(0)^(1) (1)/(sqrt(1+x)-sqrt(x))dx` |
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Answer» `underset(0)overset(1)int (1)/(sqrt(1+x)-sqrt(x))dx` `=underset(0)overset(1)int((1)/(sqrt(1+x)-sqrtx)xx(sqrt(1+x)+sqrtx)/(sqrt(1+x)+sqrtx))dx` `=underset(0)overset(1)int (sqrt(1+x)+sqrtx)/((1+x)-x)dx` `=underset(0)overset(1)int (sqrt(1+x)+sqrtx)dx` `=[2/3(1+x)^(3//2)-2/3x^(3//2)]_(0)^(1)` `=[2/3(1+1)^(3//2)-2/3(1)^(3//2)] -[2/3(1+0)^(3//2)-2/3(0)^(3//2)]` `=2/3(2)^(3//2)-2/3-2/3=2/3[2sqrt2-1-1]` `=2/3(2sqrt2-2)=4/3(sqrt2-1)` |
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| 144. |
मूल्यांकन कीजिएः `int_(0)^(pi) sin^(3)xdx` |
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Answer» `underset(0)overset(pi)int sin^(3)xdx` `underset(0)overset(pi)int (3sin x-sin 3x)/(4)dx, [therefore sin 3x=3sin x-4sin^(3)x]` `=1/4underset(0)oversetpiint (3 sin x- sin 3x)dx` `=1/4[-3 cosx+(cos 3x)/(3)]_(0)^(pi)` `=1/4[{-3 cos pi+(cos 3x)/(3)} -{-3 cos 0+(cos 0)/(3)}` `=1/4[{-3xx(-1)+((-1))/(3)}-{-3+1/3}]` `=1/4[(3-1/3)-(-3+1/3)]` `=1/4 (6-2/3)=1/4xx16/3=4/3` |
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| 145. |
मूल्यांकन कीजिएः `int_(0)^(pi//2) cos^(3)xdx` |
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Answer» `underset(0)overset(pi//2)int cos^(3)xdx` `=underset(0)overset(pi//2)int ((cos 3x+3cosx)/(4))dx [therefore cos 3x=4cos^(3)x-3cosx]` `=1/4 underset(0)overset(pi//2)int (cos 3x+3cos x)dx` `=1/4 [(sin 3x)/(3)+ sin x]_(0)^(pi//2)` `=1/4[{1/3sin((3pi)/(2))+3 sin((pi)/(2))}-{(sin 0)/(3)+3 sin 0}]` `=1/4 [{1/3 xx (-1)+3}-(0+0)]` `=1/4(-1/3+3)=1/4xx8/3=2/3` |
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| 146. |
`lim_(nrarroo) [(1)/(n^(2))sec^(2).(1)/(n^(2))+(2)/(n^(2))sec^(2).(4)/(n^(2))+…+(n)/(n^(2))sec^(2)1]` का मान है -A. `(1)/(2)tan1`B. `tan1`C. `(1)/(2)"cosec 1"`D. `(1)/(2)sec1` |
| Answer» Correct Answer - A | |
| 147. |
`int_(ln 3)^(ln 4)(e^(x)sqrt(e^(x)-3))/(e^(x)-2)dx` का मान है -A. `(4-pi)/(2)`B. `4-(pi)/(2)`C. `2-pi`D. `(2-pi)/(2)` |
| Answer» Correct Answer - A | |
| 148. |
`int_(-1)^(3)[tan^(-1)((x)/(x^(2)+1))+tan^(-1)((x^(2)+1)/(x))]dx` का मान है-A. `2pi`B. `pi`C. `(pi)/(2)`D. `(pi)/(4)` |
| Answer» Correct Answer - A | |
| 149. |
`int_(0)^(pi//2)sin 2x tan^(-1)(sin x)dx` का मान होगा -A. `(pi)/(2)-1`B. `(pi)/(2)+1`C. `(pi)/(3)-1`D. `(pi)/(3)+1` |
| Answer» Correct Answer - A | |
| 150. |
निम्न फलनों के मान ज्ञात कीजिये - `int_(pi//4)^(pi//3)sec^(2)xdx` |
| Answer» Correct Answer - `sqrt3-1` | |