Question

$\underset{0}{\overset{\infty }{\int }}\frac{x^{3}dx}{{\left(1+x^2\right)}^{9/2}}$ is equal to

• A. $\frac{2}{35}$
• B. $\frac{4}{35}$
• C. $\frac{6}{35}$
• D. $\frac{3}{35}$

Answer 1

Answer: (A)

Put $x=\tan \theta \Rightarrow dx={\sec }^2\theta d\theta$
$\Rightarrow \underset{0}{\overset{\infty }{\int }}\frac{x^3}{{\left(1+x^2\right)}^{9/2}}dx=\underset{0}{\overset{\pi /2}{\int }}\frac{{\tan }^3\theta }{{\sec }^9\theta }\cdot {\sec }^2\theta d\theta$
$=\underset{0}{\overset{\pi /2}{\int }}{\sin }^3\theta {\cos }^4\theta d\theta =\frac{\mathrm{2.3.1}}{\mathrm{7.5.3.1}}=\frac{2}{35}$