Advertisements
Advertisements
प्रश्न
Evaluate the integral by using substitution.
`int_0^1 sin^(-1) ((2x)/(1+ x^2)) dx`
Advertisements
उत्तर
Let `int_0^1 sin^-1 ((2x)/(1 + x^2)) dx`
Substituting x = tan θ
`dx = sec^2 theta d theta`
And `(2 tan theta)/(1 + tan^2 theta) = sin 2 theta`
When x = 0
⇒ θ = 0
or x = 1
`=> theta = pi/4`
Hence, `int_0^(pi/4) sin^-1 (sin 2 theta) xx sec^2 theta d theta`
`2 = int_0^(pi/4) theta sec^2 theta d theta`
`= 2 [(theta . tan theta)_0^(pi/4) - int_0^(pi/4) 1 * tan theta d theta]`
`= 2 [pi/4 tan pi/4 - 0] - 2 [log cos theta]_0^(pi/4)`
`= pi/4 + 2 [log cos pi/4 - log cos 0]`
`= pi/2 + 2 [log 1/sqrt2 - log 1]`
`= pi/2 - log 2`
APPEARS IN
संबंधित प्रश्न
Evaluate : `int1/(3+5cosx)dx`
Evaluate `∫_0^(3/2)|x cosπx|dx`
Evaluate :
`∫_(-pi)^pi (cos ax−sin bx)^2 dx`
find `∫_2^4 x/(x^2 + 1)dx`
Evaluate :
`∫_0^π(4x sin x)/(1+cos^2 x) dx`
Evaluate :
`int_e^(e^2) dx/(xlogx)`
Evaluate of the following integral:
(i) \[\int x^4 dx\]
Evaluate of the following integral:
Evaluate of the following integral:
Evaluate:
Evaluate the following definite integral:
Evaluate the following integral:
\[\int\limits_0^2 \left| x^2 - 3x + 2 \right| dx\]
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate each of the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Find : \[\int e^{2x} \sin \left( 3x + 1 \right) dx\] .
Evaluate: `int_-π^π (1 - "x"^2) sin "x" cos^2 "x" d"x"`.
`int_0^1 x(1 - x)^5 "dx" =` ______.
`int_0^1 sin^-1 ((2x)/(1 + x^2))"d"x` = ______.
`int_0^1 x^2e^x dx` = ______.
The value of `int_0^1 (x^4(1 - x)^4)/(1 + x^2) dx` is
Evaluate:
`int (1 + cosx)/(sin^2x)dx`
If `int x^5 cos (x^6)dx = k sin (x^6) + C`, find the value of k.
If \[t=\tan^{-1}x\], what is \[dt\]?
After using \[t=\tan^{-1}x\] in \[\int_{0}^{1}\frac{\tan^{-1}x}{1+x^2}\,dx\], which integral results?
What is the value of \[\int_{0}^{1}\frac{\tan^{-1}x}{1+x^2}\,dx\]?
When an integral is continued in the new variable, what must be done to its limits?
