Question

Evaluate the integral
${\int }_0^{\pi /4}({\tan }^{n}x+{\tan }^{n-2}x)\frac{d}{dx}\left(3\right\{x\}-2[x{]}^2-3\left[x{]}^3\right)dx$

• A. $\frac{1}{n-1}$
• B. $\frac{2}{n-2}$
• C. $\frac{3}{n-1}$
• D. $\frac{3}{n-2}$

Answer 1

The correct option is C $\frac{3}{n-1}$

$3{\int }_0^{\pi /4}({\tan }^{n}x+{\tan }^{n-2}x)dx$
$3{\int }_0^{\pi /4}{\tan }^{n-2}x\left({\sec }^{2}x\right)dx$
$=3{\int }_0^{\pi /4}{\tan }^{n-2}xd\left(\tan x\right)$
$=3\frac{{\tan }^{n-1}}{n-1}{|}_0^{\pi /4}$
$=3\left(\frac{1}{n-1}\right)$
$=\left(\frac{3}{n-1}\right)$