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InterviewSolution
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.
| 1. |
(i) `int(6x^(5) - 2/(x^(4)) - 7x + 3/x - 5 + 4e^(x) + 7^(x))dx` (ii) `int (8-x+2x^(3) -6/(x^(3)) +2x^(-5) + 5x^(-1)) dx` , (iii) `int (x/a+a/x+x^(a) +a^(x) + ax) dx` |
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Answer» Correct Answer - (i) `x^(6) + (2)/(3x^(3)) - (7x^(2))/(2) + 3|log| - 5x + 4e^(x) + (7^(x))/(log 7) + C` (ii) `8x - (x^(2))/(2) +(x^(4))/(2) + (3)/(x^(2)) - (1)/(2x^(4)) + 5 log |x| + C` (iii) `(x^(2))/(2a) + a log |x| +(x^((a+1)))/((a+1)) +(a^(2))/(log a ) +(ax^(2))/(2) + C` |
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| 2. |
Evaluate `int(secx)/((secx+tanx)) dx`. |
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Answer» `int(sec x)/((secx + tanx)) dx = int(secx)/((secx + tanx)) xx((secx - tanx))/((secx - tanx)) dx` `= int ((sec^(2)x - secx tanx))/((sec^(2)x - tan^(2)x)) dx` `= int(sec^(2)x - secx tanx) dx` `= int sec^(2) xdx - int sec x tan x dx = tan x - secx + C`. |
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| 3. |
Evaluate `int ((cosx - 2cos 2alpha)/(cosx - cos alpha)) dx`. |
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Answer» `int((cos 2x-cos2alpha)/(cosx - cos alpha)) dx = int((2cos^(2)x - 1) - (2cos^(2)alpha-1))/((cosx - cos alpha)) dx` `= 2int((cos^(2)x - cos^(2)alpha))/((cosx - cosalpha)) dx = 2 int(cosx+cosalpha) dx` `= 2 intcosx dx + 2 cos alpha. intdx = 2 sinx +2x cos alpha + C`. |
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| 4. |
Evaluate : (i) `int((4-5 cosx)/(sin^(2)x))dx` , (ii) `int((1-cos2x)/(1+cos2x)) dx` (iii) `int (1)/(sin^(2)x cos^(2)x) dx` , (iv) `int(cos2x)/(cos^(2)x sin^(2) x) dx` |
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Answer» (i) `int((4-5cosx)/(sin^(2)x))dx = int(4/(sin^(2)x) - (5cosx)/(sin^(2)x))dx` `= int(4cosec^(2) x - 5 cosec x cot x) dx` `= 4 intcosec^(2) xdx - 5 int cosec x cot x dx` `= 4 (-cotx) - 5(-cosecx) + C` `= - 4 cot x + 5 cosec x + C`. (ii) `int((1-cos 2x)/(1+ cos2x)) dx = int(2sin^(2)x)/(2 cos^(2)x) dx = inttan^(2)x dx` `= int(sec^(2)x - 1)dx = intsec^(2)x dx - intdx` `= tanx-x + C`. (iii) `int (1)/(sin^(2)x cos^(2)x) dx = int((sin^(2)x +cos^(2)x)/(sin^(2)xcos^(2)x))dx` `= int(1/(cos^(2)x) + 1/(sin^(2)x)) dx` `= intsec^(2) xdx + intcosec^(2)xdx = tan x - cot x +C`. (iv) `int(cos2x)/(cos^(2)x sin^(2)x) dx = int((cos^(2)x - sin^(2)x)/(cos^(2)x sin^(2)x)) dx` `= int((1)/(sin^(2)x) - (1)/(cos^(2)x))dx` `= intcosec^(2)x dx - int sec^(2)x dx = -cotx - tan x+ C`. |
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| 5. |
(i) `int(1)/((1+cos2x))dx` , (ii) `int (1)/((1-cos2x)) dx` |
| Answer» Correct Answer - (i) `1/2tanx + C`, (ii) `- 1/2 cot x + C` | |
| 6. |
`int sqrt(1+sin2x)dx` |
| Answer» Correct Answer - `sinx - cos x + C` | |
| 7. |
`int tan^(-1) ((sin2x)/(1+cos 2x))dx` |
| Answer» Correct Answer - `(x^(2))/(2) + C` | |
| 8. |
`intsqrt[1+sin2x]dx`A. `sinx + cosx + C`B. `-sinx + cos x + C`C. `sin x - cos x + C`D. `-sin x - cos x + C` |
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Answer» Correct Answer - C `(1+ sin 2x) = sin^(2) x + cos^(2)x + 2sin x cos x = (sinx +cosx)^(2)` `sqrt(1+sin2x) = (sinx + cosx)`. |
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| 9. |
Evaluate : `intsqrt(1-sin2x)dx`. |
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Answer» `intsqrt(1-sin2x) dx = int(cos^(2) x + sin^(2)x - 2 sin x cos x)^(1//2) dx` `= intsqrt((cosx -sinx)^(2))dx` `= int(cosx - sinx) dx = intcos x dx - int sin xdx` `= sinx - (-cosx) + C = sin x + cos x + C`. |
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| 10. |
Evaluate : (i) `int3x^(2)dx` , (ii) `int2^((x+3)) dx` |
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Answer» (i) `int3x^(2)dx = 3intx^(2)dx = 3.(x^(2))/(3)+C=x^(3)+C`. (ii) `int2^((x+3)) dx = int2^(x).2^(3)dx = 8int2^(x)dx = 8.(2^(x))/(" log" 2) + C = (2^((x+3)))/(" log" 2)+ C = (2^((x+3)))/(" log" 2) + C`. |
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| 11. |
`int 3sqrt(x) dx = ?`A. `5/3 x^(5/3) + C`B. `3/5 x^(5/3) + C`C. `5/3 x^(3/5) + C`D. `3/5 x^(3/5) + C` |
| Answer» Correct Answer - B | |
| 12. |
`int cos^(-1) (sinx) dx` |
| Answer» Correct Answer - `((pix)/(2)- (x^(2))/(2) + C)` | |
| 13. |
`int 1/(3sqrt(x)) dx = ?`A. `3/2 x^(2/3) + C`B. `3/(2x^(2/3)) + C`C. `2/(3^(2/3)) + C`D. `2/3x^(3/2) + C` |
| Answer» Correct Answer - A | |
| 14. |
`int3^(x) dx = ?`A. `3^(x) (log3) + C`B. `3^(x) + C`C. `(3^(x))/(log 3) + C`D. `(log3)/(3^(x)) + C` |
| Answer» Correct Answer - C | |
| 15. |
Evaluate: `inttan^(-1){sqrt(((1-sinx)/(1+sinx)))} dx , -pi//2 |
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Answer» Correct Answer - `(pix)/(4) - (x^(2))/(4) + C` `I = int tan^(-1) sqrt((1-cos(pi/2 - x))/(1+ cos(pi/2-x))) dx` `= inttan^(-1) sqrt((2sin^(2)(pi/4-x/2))/(2cos^(2)(pi/4-x/2)))dx = inttan^(-1){tan(pi/4 - x/2)} dx` `= int (pi/4- x/2) dx = (pix)/(4) - (x^(2))/(4) + C`. |
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| 16. |
`int 3sqrt(x) dx = ?`A. `3/4x^(3/4) + C`B. `4/3x^(3/4) + C`C. `3/4 x^(4/3) +C`D. `4/3 x^(4/3) +C` |
| Answer» Correct Answer - C | |
| 17. |
`int(1)/(sin^(2)x cos^(2)x) dx = ?`A. `tanx + cot x + C`B. `-tan x + cot x + C`C. `tanx - cot x + C`D. none of these |
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Answer» Correct Answer - C Write `1 = (sin^(2)x + cos^(2)x)` |
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| 18. |
(i) `int(x^(2) - 1/(x^(2)))^(3) dx` , (ii) `int(sqrt(x) -1/(sqrt(x))) dx` (iii) `int(sqrt(x) + 1/(sqrt(x)))^(2) dx` , (iv) `int((1+2x)^(3))/(x^(4))` (v) `int((1+x)^(3))/(sqrt(x)) dx` , (vi) `int(2x^(2)+x-2)/((x-2)) dx` |
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Answer» Correct Answer - (i) `(x^(7))/(7) +1/(5x^(5)) - x^(3) - 3/x + C`, (ii) `2/3 x^(3//2) - 2x^(1//2) +C` `(x^(2))/(2) + log |x| + 2x + C` , (iv) `- 1/(3x^(3)) + 8 log |x| - (3)/(x^(2)) - (12)/(x) +C` (v) `2sqrt(x) + 2/7 x^(7//2) +2x^(3//2) + 6/5 x^(5//2) + C` (vi) `x^(2) + 5x + 8log|x-2| + C` |
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| 19. |
`int (cos(x+a))/(sin(x+b))dx` |
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Answer» Correct Answer - `cos (a-b) log|sin(a+b)|-xsin(a-alpha)|+C` `I = int(cos(x+b+a-b))/(sin(x+b))dx` `= int(cos(x+b) cos(a-b)-sin(x+b)sin(a-b))/(sin(x+b)) dx` `= cos(a-b)intcot(x+b)dx-sin(a-b)intdx`. |
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| 20. |
`int1/(sqrt(x+3)-sqrt(x+2)) dx` |
| Answer» Correct Answer - `2/3(x+3)^(3//2) + 2/3(x+2)^(3//2) + C` | |
| 21. |
`int(1)/(x^(3))dx = ?`A. `(-3)/(x^(2)) + C`B. `(-1)/(2x^(2)) + C`C. `(-1)/(3x^(2)) + C`D. `(x^(-2))/(2) + C` |
| Answer» Correct Answer - B | |
| 22. |
`int[1+(1)/((1+x^(2)))-(2)/(sqrt(1-x^(2)))+(5)/(x sqrt(x^(2)-1))+a^(x)]dx` |
| Answer» Correct Answer - `x + tan^(-1)x - 2sin^(-1)x + 5sec^(-1) x + (a^(x))/(log a) + C`. | |
| 23. |
`int (cos 2x)/(cosx^(2) sin^(2)x) dx = ?`A. `-cotx - tan x + C`B. `- cot x + tanx + C`C. `cot x - tan x + C`D. `cot x + tanx + C` |
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Answer» Correct Answer - A `cos 2x = (cos^(2)x - sin^(2)x)`. |
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| 24. |
`int(sin(x-alpha))/(sin(x+alpha)) dx` |
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Answer» Correct Answer - `xcos 2alpha-sin 2alpha.log|sin(x+alpha)|+C` `I = int(sin(x+alpha-2alpha))/(sin(x+alpha)) dx`. ` = int(sin(x+alpha - 2alpha))/(sin(x+alpha)) dx` `= int(sin(x+alpha)cos2alpha-cos(x+alpha)sin2alpha)/(sin(x+alpha)) dx = cos 2alpha int dx - sin 2alpha . int(cos(x+alpha))/(sin(x+alpha))` `= x cos 2alpha - sin 2alpha. log|sin(x+alpha)| + C`. |
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| 25. |
`inttan^(-1){sqrt((1-cos2x)/(1+cos2x))} dx = ?`A. `1/((1+x^(2))) + C`B. `1/(sqrt(1+x^(2))) + C`C. `1/(sqrt(1-x^(2))) + C`D. `(x^(2))/(2) + C` |
| Answer» Correct Answer - D | |
| 26. |
(i) `int((x^(2) - 1)/(x^(2) + 1))dx` , (ii) `int ((x^(6)- 1)/(x^(2) + 1))dx` (iii) `int ((x^(4))/(1+x^(2)))dx` , (iv) `int((x^(2))/(1+x^(2)))dx` |
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Answer» Correct Answer - (i) `x - 2tan^(-1)x + C` , (ii) `(x^(5))/(5) +(x^(3))/(3) + x - 2tan^(-1) x + C` (iii) `(x^(3))/(3) - x + tan^(-1)x + C` , (iv) `x - tan^(-1) + C`. |
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| 27. |
`int 2^(logx) dx = ?`A. `(2^(logx+1))/((logx+1)) + C`B. `(x^((log2+1)))/((log2+1)) + C`C. `(2^(logx))/(log 2) + C`D. `(2^(log x))/(2) + C` |
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Answer» Correct Answer - B `2^(log x ) = x^(log 2)` |
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| 28. |
`int(dx)/(sqrt(x+1)+sqrt(x+2))` |
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Answer» Correct Answer - `2/3 (x+2)^(3//2) - 2/3(x+1)^(3//2) + C`. `I = int(1)/((sqrt(x+2)+ sqrt(x+1))) xx ((sqrt(x+2) - sqrt(x+1)))/((sqrt(x+2) - sqrt(x+1))) dx` `= intsqrt(x+2) dx - intsqrt(x+1) dx =2/3 (x+2)^(3//2) - 2/3 (x+1)^(3//2) +C`. |
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| 29. |
`int ((1+tanx)/(1-tanx))dx` |
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Answer» Correct Answer - `-log |cosx 0 sinx| + C` `I = int((1+(sinx)/(cosx))/(1-(sinx)/(cosx)))dx = int((cosx+sinx))/((cosx-sinx)) dx` `= -int(dt)/(t)`, where `(cosx -sinx) =t` `= - log|t| +C =-log|cosx-sinx|+C`. |
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| 30. |
`int sin^(-1) (cosx) dx = ?`A. `cosec x + C`B. `(pix)/(2) +(x^(2))/(2) + C`C. `(pix)/(2) - (x^(2))/(2) + C`D. `(x^(2))/(2) - (pix)/(2) + C` |
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Answer» Correct Answer - C `sin^(-1) (cosx) = sin^(-1) {sin(pi/2 - x)} = (pi/2 - x)`. |
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| 31. |
`int 1/(1+cosx) dx=`A. `-cot x + cosec x + C`B. `cot x - cosec x + C`C. `cot x + cosec x + C`D. none of these |
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Answer» Correct Answer - A `I = int{(1)/((1+cosx)) xx ((1-cosx))/((1-cosx))} dx = int((1-cosx))/(sin^(2)x) dx` `= {(-1)/(sin^(2)x) - (cosx)/(sin^(2)x)} dx = int(cosec^(2)x - cosec x cot x ) dx`. |
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| 32. |
`intx^(5/3)dx = ?`A. `3/5x^(2/3) + C`B. `8/3 x^(8/3) +C`C. `3/8 x^(8/3)+ C`D. `5/3x^(8/3) + C` |
| Answer» Correct Answer - C | |
| 33. |
`int(1-x)sqrt(x) dx` |
| Answer» Correct Answer - `2/15xsqrt(x) (5-3x) + C`. | |
| 34. |
`int cosecx(cosec x + cot x) dx = ?`A. `cot x - cosec x + C`B. `-cot x + cosec x + C`C. `cotx + cosec x + C`D. `-cot x - cosec + C` |
| Answer» Correct Answer - D | |
| 35. |
`int(3sinx +4 cosecx)^(2) dx` |
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Answer» Correct Answer - `(57)/(2) x - 9/4sin 2x - 16 cot x + C` `I = int (9sin^(2)x + 16cosec^(2)x + 24)dx` `= int{9((1-cos2x)/(2)) + 16 cosec^(2)x + 24} dx`. `= int(57/2- 9/2 cos 2x + 16 cosec^(2)x) dx`. |
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| 36. |
`int (sec^(2)x)/(cosec^(2)x) dx` |
| Answer» Correct Answer - `tan x -x + C` | |
| 37. |
`intx^(6) dx = ?`A. `7x^(7) +C`B. `(x^(7))/(7) + C`C. `6x^(5) + C`D. `6x^(7) + C` |
| Answer» Correct Answer - B | |
| 38. |
`int (sec^(2) x)/(" cosec "^(2)x) dx`A. `tanx + x + C`B. `tanx - x + C`C. `-tanx + x + C`D. `- tanx + x + C` |
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Answer» Correct Answer - B `(sec^(2)x)/(cosec^(2)x) = (sin^(2)x)/(cos^(2)x) = tan^(2)x = (sec^(2) x - 1)`. |
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| 39. |
`int cot^(2) x dx = ?`A. `-cot x -x + C`B. `cot x-x + C`C. `-cotx + x + C`D. `cot x + x + C` |
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Answer» Correct Answer - B `cot^(2)x = (cosec^(2)x - 1)`. |
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| 40. |
`int (cot x)/((cosec x - cot x)) dx = ?`A. `-cosec x - cot x - x + C`B. `cosec x - cot x - x + C`C. `-cosec x + cot x -x + C`D. `cosec x + cot x -x + C` |
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Answer» Correct Answer - A `I = int{(cotx)/((cosecx - cot))xx ((cosecx + cotx))/((cosec x + cotx))} dx` `= int(cosecx cot x + cot^(2)x) dx = int(cosecx cotx +cosec^(2)x - 1) dx`. |
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| 41. |
`int ((1+ sinx))/((1- sinx)) dx=?`A. `2 sin x + 2 sec x + x + C`B. `2 tan x + 2 sec x - x + C`C. `tanx + sec x - x + C`D. none of these |
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Answer» Correct Answer - B `I = int{((1+sinx))/((1-sinx)) xx((1-sinx))/((1+sinx))} dx = int((1+sinx)^(2))/((1-sinx^(2)x)) dx` `= int((1+sin^(2)x+ 2sinx))/(cos^(2)x) dx = int(sec^(2)x + tan^(2)x +2sec xtanx) dx` `= int (2sec^(2) - 1+ 2 sec x tanx ) dx`. |
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| 42. |
`int(sinx)/(1+sin x) dx`A. `sec x +tanx + x + C`B. `sec x - tanx + x + C`C. `-sec x + tanx + x + C`D. none of these |
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Answer» Correct Answer - B `I = int{(sinx)/((1+ sinx)) xx ((1-sinx))/((1-sinx))}dx = int(sinx(1-sinx))/((1-sin^(2)x)) dx` `= int ((sinx - sin^(2)x))/(cos^(2)x) dx = int((sinx)/(cos^(2)x) - tan^(2)x)dx` `= int (sec x tan x - sec^(2)x +1) dx`. |
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| 43. |
`int ((1- sinx))/(cos^(2)x) dx= ?`A. `tanx +sec x + C`B. `tanx - sec x + C`C. `-tanx + sec x + C`D. `-tanx - sec x + C` |
| Answer» Correct Answer - B | |
| 44. |
`int {(2-3 sinx)/(cos^(2)x) } dx` |
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Answer» Correct Answer - `2tan x - 3 sec x + C` `I = int{(2)/(cos^(2)x) - (3sinx)/(cos^(2)x)} dx = int(2sec^(2)x - 3 secxtanx) dx`. |
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| 45. |
`int sin 3x sin 2x dx = ?`A. `- 1/5 cos 5x + C`B. `1/2 sin x + 1/10 sin 5x - C`C. `1/2 sinx - 1/10 sin 5x - C`D. `-1/3 cos 3 x - 1/2 sin 2 x + C` |
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Answer» Correct Answer - C `I = 1/2 int2sin 3xsin2x dx` `= 1/2(cosx -cos5x) dx = 1/2 sin x - 1/10sin 5x + C`. |
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| 46. |
`int(sinx)/(sin(x-alpha)) dx = ?`A. `x cos alpha + (sin alpha) log |sin (x-alpha)| + C`B. `x sin alpha + (sin alpha) log |sin (x-alpha) | + C`C. `x cos alpha - (sin alpha) log |sin(x-alpha) | + C`D. `x sin alpha - (sin alpha) log |sin (x-alpha)| + C` |
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Answer» Correct Answer - A `I = (sin(x-alpha +alpha))/(sin(x-a)) dx = int(sin(x-alpha) cos alpha +cos(x-alpha)sinalpha)/(sin(x-alpha)) dx` `= (cosalpha) int dx +(sinalpha) int cot (x- alpha) dx`. `= xcosalpha+(sinalpha) log|sin(x-alpha) | +C`. |
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| 47. |
`int sin^(-1)((2tanx )/(1+tan^(2)x))dx = ?`A. `-x^(2) + C`B. `x^(2) + C`C. `(x^(2))/(2) + C`D. `2x^(2) + C` |
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Answer» Correct Answer - B `sin^(-1)((2tanx)/(1+tan^(2)x))= sin^(-1) (sin 2x) = 2x`. |
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| 48. |
`int (sin^3x+cos^3x)/(sin^2x*cos^2x)dx`A. `sinx - cos x + C`B. `tanx - cos x + C`C. `sec x - cosec + X`D. none of these |
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Answer» Correct Answer - C `I = int (sin^(3)x)/(sin^(2)xcos^(2)x) dx +int(cos^(2)x)/(sin^(2)xcos^(2)x) dx`. `int secx tanx dx +int cosec xcot x dx` `= sec x - cosec x+ C`. |
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| 49. |
`int(a x+b)/(c x+d) dx`A. `(ax)/(c ) +log|cx +d| + C`B. `a/c + log|cx + d| + C`C. `(ax)/(c ) + ((bc- ad))/(c^(2)) log |cx + d| + C`D. none of these |
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Answer» Correct Answer - C On dividing `(ax+b)` by `(cx +d)`, we get `int((ax + b))/((cx + d)) = int {a/c +((bc-ad))/(c(cx+d))} dx` `= inta/c dx + ((bc-ad))/(c^(2)) .int(c )/((cx+d)) dx` `= (ax)/(c ) + ((bc - ad))/(c^(2)) log|cx+d| + C`. |
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| 50. |
`int (((x^(4) + 1))/((x^(2) + 1))) dx = ?`A. `(x^(3))/(3) + x - tan^(-1) x + C`B. `(x^(3))/(3) - x + 2 tan^(-1) x + C`C. `(x^(3))/(3)0 + x - 2 tan^(-1) x +C`D. none of these |
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Answer» Correct Answer - B On dividing `(x^(4) + 1)` by `(x^(2) +1)`, we get `((x^(4)+1))/((x^(2) + 1)) = (x^(2) - 1) + 2/((1+x^(2)))`. |
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