Evaluate `int(3sinx+2cosx)/(3cosx+2sinx)dx.`

1.	Evaluate `int(3sinx+2cosx)/(3cosx+2sinx)dx.`
Answer» We have `I=int(3sinx+2cosx)/(3cosx+2sinx)dx` Let `3sinx+2cosx=mu(d)/(dx)(3cosx+2sinx)+lambda(3cosx+2sinx)` `=mu(-3sinx+2cosx)+lambda(3cosx+2sinx)` Comparing the coefficients of `sin x` and `cosx` on both sides, we get `-3mu+2lambda=3 " and " 2mu+3lambda=2` ` " or " lambda=(12)/(13) " and " mu=-(5)/(13) ` `:. I=int(mu(-3sinx+2cosx)+lambda(3cosx+2sinx))/(3cosx+2sinx)dx` `=lambda int 1dx+mu int(-3sinx+2cosx)/(3cosx+2sinx)dx` `=lambda x +mu int(dt)/(t), " where " t=3cosx+2sinx` `=lambda x +mu log\|t\|+C` `=(12)/(13)x+(-5)/(13)log\|3cosx+2sinx\|+C`

Discussion

No Comment Found

Related InterviewSolutions

Find:Integrate ((x4 - 1)1/4 )/(x6) dx\(\int\frac{{x^4-1}^{1/4}dx}{x^6}\)
`int((2+secx)secx)/((1+2secx)^2)dx=`A. `(1)/("2 cosec x"+cotx)+C`B. `"2 cosec x"+cotx+C`C. `(1)/("2 cosec x"-cotx)+C`D. `"2 cosec x"-cotx+C`
If `int(e^(4x)-1)/(e^(2x))log((e^(2x)+1)/(e^(2x-1)))dx=(t^(2))/(2)logt-(t^(2))/(4)-(u^(2))/(2)logu+(u^(2))/(4)+C,` thenA. `u=e^(x)+e^(-x)`B. `u=e^(x)-e^(-x)`C. `t=e^(x)+e^(-x)`D. `t=e^(x)-e^(-x)`
Let F(x) be an indefinite integral of `sin^(2)x` Statement-1: The function F(x) satisfies `F(x+pi)=F(x)` for all real x. because Statement-2: `sin^(3)(x+pi)=sin^(2)x` for all real x. A) Statement-1: True , statement-2 is true, Statement -2 is not a correct explanation for statement -1 c) Statement-1 is True, Statement -2 is False. D) Statement-1 is False, Statement-2 is True.
Let F(x) be an indefinite integral of `sin^(2)x` Statement-1: The function F(x) satisfies `F(x+pi)=F(x)` for all real x. because Statement-2: `sin^(3)(x+pi)=sin^(2)x` for all real x. A) Statement-1: True , statement-2 is true, Statement -2 is not a correct explanation for statement -1 c) Statement-1 is True, Statement -2 is False. D) Statement-1 is False, Statement-2 is True.A. Both the Statement are true and Statement 2 is the correct explanation of Statement 1B. Both the Statement are true but Statement 2 is not the correct explanation of Statement 1C. Statement 1 is true but Statement 2 is falseD. Statement 1 is false but Statement 2 is true
`int(x^3-x)/(1+x^6)dx` is equal toA. `(1)/(6)log.(x^(4)-x^(2)+1)/(x(x^(2)+1))+C`B. `(1)/(6)tan^(-1).((x^(2)+1)^(2))/(2)+C`C. `log.(x^(4)-x^(2)+1)/((1+x^(2))^(2))+C`D. `tan^(-1).((x^(2)+1)^(2))/(2)+C`
The integral `int(dx)/(x^(2)(x^(4)+1)^(3//4))` equalA. `(x^(4)+1)/(x^(4))^(1//4)+C`B. `(x^(4)+1)^(1//4)+C`C. `-(x^(4)+1)^(1//4)+C`D. `-(x^(4)+1)/(x^(4))+C`
`"The integral " int(dx)/(x^(2)(x^(4)+1)^(3//4))" equals"`A. `((x^(4)+1)/(x^(4)))^(1//4)+c`B. `(x^(4)+1)^(1//4)+c`C. `-(x^(4)+1)^(1//4)+c`D. `-((x^(4)+1)/(x^(4)))^(1//4)+c`
The integral `int[2x^[12]+5x^9]/[x^5+x^3+1]^3.dx` is equal to- (A) `x^10 / (2(x^5 + x^3 +1)^2) ` (B) `x^5/ (2(x^5 + x^3 +1)^2) ` (C) `-x^10 / (2(x^5 + x^3 +1)^2) ` (D) `- x^5 / (2(x^5 + x^3 +1)^2) `A. `(x^(10))/(2(x^(5)+x^(3)+1)^(2))+C`B. `(x^(5))/(2(x^(5)+x^(3)+1)^(2))+C`C. `(-x^(10))/(2(x^(5)+x^(3)+1)^(2))`D. `(-x^(5))/((x^(5)+x^(3)+1)^(2))+C`
The integral `int[2x^[12]+5x^9]/[x^5+x^3+1]^3.dx` is equal to- (A) `x^10 / (2(x^5 + x^3 +1)^2) ` (B) `x^5/ (2(x^5 + x^3 +1)^2) ` (C) `-x^10 / (2(x^5 + x^3 +1)^2) ` (D) `- x^5 / (2(x^5 + x^3 +1)^2) `A. `(x^(10))/(2(x^(5)+x^(3)+1)^(2))+C`B. `x^(5)/(2(x^(5)+x^(3)+1)^(2))+C`C. `-x^(10)/(2(x^(5)+x^(3)+1)^(2))+C`D. `-x^(5)/(x^(5)+x^(3)+1)^(2)+C`

Evaluate `int(3sinx+2cosx)/(3cosx+2sinx)dx.`

Discussion

No Comment Found

Related InterviewSolutions

Reply to Comment