Advertisements
Advertisements
Question
Evaluate the integral by using substitution.
`int_0^(pi/2) (sin x)/(1+ cos^2 x) dx`
Advertisements
Solution
`int_0^(pi/2) (sin x)/(1 + cos^2 x) ` dx
Substituting cos x = t,
⇒ - sin x dx = dt
And x = 0, t = 1, x `= pi/2,` t = 0
Hence, `I = - int_1^0 1/(1 + t^2)` dt
`= - [tan^-1 t]_1^0`
`= - [tan^-1 0 - tan^-1 1]`
`= - [0 - pi/4]`
`= pi/4`
APPEARS IN
RELATED QUESTIONS
Evaluate: `int (1+logx)/(x(2+logx)(3+logx))dx`
Evaluate `∫_0^(3/2)|x cosπx|dx`
find `∫_2^4 x/(x^2 + 1)dx`
Evaluate :
`∫_0^π(4x sin x)/(1+cos^2 x) dx`
Evaluate: `intsinsqrtx/sqrtxdx`
Evaluate the integral by using substitution.
`int_0^1 x/(x^2 +1)`dx
Evaluate the integral by using substitution.
`int_1^2 (1/x- 1/(2x^2))e^(2x) dx`
The value of the integral `int_(1/3)^4 ((x- x^3)^(1/3))/x^4` dx is ______.
Evaluate `int_0^(pi/4) (sinx + cosx)/(16 + 9sin2x) dx`
Evaluate:
Evaluate:
Evaluate the following definite integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate each of the following integral:
Evaluate each of the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate: \[\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x}dx\] .
`int_(pi/5)^((3pi)/10) [(tan x)/(tan x + cot x)]`dx = ?
`int_0^1 x(1 - x)^5 "dx" =` ______.
`int_0^3 1/sqrt(3x - x^2)"d"x` = ______.
`int_0^(pi4) sec^4x "d"x` = ______.
Evaluate: `int x/(x^2 + 1)"d"x`
Which limits are used to evaluate the resulting expression in Method: Resubstitution?
After using \[t=\tan^{-1}x\] in \[\int_{0}^{1}\frac{\tan^{-1}x}{1+x^2}\,dx\], which integral results?
