Advertisements
Advertisements
Question
Find `int_0^1 x(tan^-1x) "d"x`
Solution
I = `int_0^1x(tan^-1x)^2 "d"x`
Integrating by parts, we have
I = `x^2/2[(tan^-1x)^2]_0^1 - 1/2 int_0^1 x^2 * 2 (tan^-1x)/(1 + x^2) "d"x`
= `pi^2/32 - int_0^1 x^2/(1 + x) * tan^-1 x"d"x`
= `pi^2/32 - 1_1`, where I1 = `int_0^1 x^2/(1 + x^2) tan^-1 x"d"x`
Now I1 = `int_0^1 (x^2 + 1 - 1)/(1 + x^2) tan^-1x "d"x`
= `int_0^1 tan^-1 x"d"x - int_0^1 1/(1 + x^2) tan^-1 x"d"x`
= `"I"_2 - 1/2 ((tan^-1x)^2)_0^1`
= `"I"_2 - pi^2/32`
Here I2 = `int_0^1 tan^-1 x"d"x = (x tan^-1x)_0^1 - int_0^1 x/(1 + x^2) "d"x`
= `pi/4 - 1/2(log|1 + x^2|)_0^1`
= `pi/4 - 1/2 log2`
Thus I2 = `pi/4 - 1/2 log 2 - pi^2/32`
Therefore, I = `pi^2/32 - pi/4 + 1/2 log2 + pi^2/32`
= `pi^2/16 - pi/4 + 1/2 log2`
= `(pi^2 - 4pi)/16 + log sqrt(2)`.
APPEARS IN
RELATED QUESTIONS
If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:
(A) 0
(B) π
(C) π/2
(D) π/4
Integrate the function in `e^x (1/x - 1/x^2)`.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following : `int cos sqrt(x).dx`
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
Choose the correct options from the given alternatives :
`int (1)/(cosx - cos^2x)*dx` =
Integrate the following w.r.t.x : cot–1 (1 – x + x2)
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int "x"^3 "e"^("x"^2)`dx
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
Solution of the equation `xdy/dx=y log y` is ______
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
Evaluate:
`int e^(ax)*cos(bx + c)dx`
Evaluate:
`inte^x sinx dx`
Prove that `int sqrt(x^2 - a^2)dx = x/2 sqrt(x^2 - a^2) - a^2/2 log(x + sqrt(x^2 - a^2)) + c`
Evaluate `int (1 + x + x^2/(2!))dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
