Question

please help me solve this maths question on integration by method of substitution......qs no. 42 ....correct answer will be marked as brainliest. ...

Answer 1

suppose
$1 + {x}^{2}$
equal to
${y}^{2}$
then solve it

Answer 2

$\displaystyle \int {\frac{\, dx}{(1+x^2)\sqrt{1+x^2}}}$

[tex]\text{Let } x = \tan\theta [/tex]

$\therefore \, dx = \sec^2\theta\,\,d\theta \, \, \, \, \, \. \text{ and }\,\,\,\, \theta=\tan^{-1}x \,\,\,\,\,\, ... (i)$

$\therefore \displaystyle \int {\frac{\sec^2\theta\,\, d\theta}{(1+\tan^2\theta)\sqrt{1+\tan^2\theta}}}$

$\displaystyle \int{\frac{\cancel{\sec^2\theta} \,\, d\theta}{\cancel{sec^2\theta}\sqrt{sec^2\theta}}}$

$\displaystyle \int{\frac{d\theta}{\sec\theta}}$

$\displaystyle \int \cos\theta \, \, d\theta$

$\sin\theta + C$

$\text{Substituting the value of }\theta\text{ from eq.(i)}$

$\sin(\tan^{-1}x)+ C$