Question

please solve this intrigration

Answer 1

Please see the attachment

Answer 2

Question :

Integrate :

1) sin inverse root x /a+x

2) x^2 dx/(x^2+a^2)(x^2+b^2)

Solution :

1) $\sf\int \sin {}^{ - 1} \sqrt{ \dfrac{x}{a + x} } \times dx$

Let $\sf \: x = a \tan { }^{2} \theta$

$\sf \implies \frac{dx}{d \theta} = 2a \times \tan \theta \times \sec {}^{2} \theta$

$\sf \implies \: dx = 2a \tan \theta \sec {}^{2} \theta \: d \theta$

put this value in integration

$\sf \int sin {}^{ - 1} \sqrt{ \dfrac{x}{a + x} } dx$

$\sf = \int \sin {}^{ - 1} \sqrt{ \dfrac{a \tan {}^{2} \theta }{a + a \tan {}^{2} \theta} } \times 2a \tan \theta \sec {}^{2} \theta d \theta$

$\sf = \int \sin {}^{ - 1} \sqrt{ \dfrac{ \cancel{a} \tan {}^{2} \theta}{ \cancel{a}(1 + \tan {}^{2} \theta)} } \times 2a \tan \theta \sec {}^{2} \theta d \theta$

$\sf = \int \sin {}^{ - 1} \sqrt{ \dfrac{ \tan \theta}{ \sec \theta} } \times 2a \tan \theta \sec {}^{2} \theta d \theta$

$\sf = \int \sin {}^{ - 1} ( \sin \theta) \times 2a \tan \theta \sec {}^{2} \theta d \theta$

$\sf = 2a \int \theta \tan \theta \sec {}^{2} \theta d\theta$

Now integrate ,by using Integration by parts

Let θ= First function and other one as second function

$\sf = 2a( \theta \int \tan \theta \sec {}^{2} \theta d\theta - \int 1\times \tan \theta \sec {}^{2} \theta \:d \theta)$

$\sf = 2a( \theta \times \dfrac{ \tan {}^{2} \theta}{2} - \int \sec {}^{2} \theta \tan \theta d \theta)$

$\sf = 2a( \theta \times \dfrac{ \tan {}^{2} \theta}{2} - \int \dfrac{\tan{}^{2}\theta}{2})$

$\sf = a( \theta \times \tan {}^{2} \theta - \int( \sec {}^{2} \theta - 1)d \theta) + c$

$\sf = a ( \theta \tan {}^{2} \theta - \tan \theta + \theta) + c$

$\sf = a( \dfrac{x}{a} \tan {}^{ - 1} \sqrt{ \dfrac{x}{a} } - \sqrt{ \dfrac{x}{a} } + \tan {}^{ - 1} \sqrt{ \dfrac{x}{a} } ) + c$

2)

$\sf \int \dfrac{x {}^{2} }{(x {}^{2} + a {}^{2} )(b {}^{2} + x {}^{2} )} dx$

$\sf = \dfrac{1}{b {}^{2} - a {}^{2} } (b \tan {}^{ - 1} ( \dfrac{x}{b} ) - a \tan {}^{ - 1} ( \dfrac{x}{a} )) + c$

for step by step Explanation refer to the attachment.