Question

The value of $\underset{0}{\overset{\frac{\pi }{4}}{\int }}2{\tan }^{3}xdx$ is

• A. $\ln 2$
• B. $\ln 2-1$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is B $\ln 2-1$

$I=\underset{0}{\overset{\frac{\pi }{4}}{\int }}2(1-{\sec }^{2}x)\tan xdx$
$I=\underset{0}{\overset{\frac{\pi }{4}}{\int }}2\tan xdx-\underset{0}{\overset{\frac{\pi }{4}}{\int }}2\tan x{\sec }^{2}xdx$
$I=2\ln |\sec x|{|}_0^{\frac{\pi }{4}}-{\tan }^{2}x{|}_0^{\frac{\pi }{4}}$
$I=2\ln |\sqrt{2}|-2\ln |1|-(1-0)$
$I=\ln 2-1$