The value of \(\int_{0}^{\pi/2}\displaystyle\frac{\tan^7 x}{\cot^7 x + \tan^7 x}\,dx\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\pi/4\)
Let
\( \displaystyle I = \int_{0}^{\pi/2} \frac{\tan^7 x}{\cot^7 x + \tan^7 x}\,dx \)
By the property \(\int_{0}^{a} f(x)\,dx = \int_{0}^{a} f(a - x)\,dx\), replace x by \(\frac{\pi}{2} - x\). Since \(\tan\left(\frac{\pi}{2} - x\right) = \cot x\) and \(\cot\left(\frac{\pi}{2} - x\right) = \tan x\):
\( \displaystyle I = \int_{0}^{\pi/2} \frac{\cot^7 x}{\tan^7 x + \cot^7 x}\,dx \)
Adding the two forms of I:
\( \displaystyle 2I = \int_{0}^{\pi/2} \frac{\tan^7 x + \cot^7 x}{\cot^7 x + \tan^7 x}\,dx = \int_{0}^{\pi/2} 1\,dx = \frac{\pi}{2} \)
\( \displaystyle I = \frac{\pi}{4} \)
NCERT Exemplar:
Integrals
