Question

The value of \[\int\limits_{- \pi/2}^{\pi/2} \left( x^3 + x \cos x + \tan^5 x + 1 \right) dx, \] is 

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Answer 1

π

\[\int_{- \frac{\pi}{2}}^\frac{\pi}{2} \left( x^3 + x\cos x + \tan^5 x + 1 \right) d x\]

\[ = \left[ \frac{x^4}{4} \right]_{- \frac{\pi}{2}}^\frac{\pi}{2} + \left[ x \sin x \right]_{- \frac{\pi}{2}}^\frac{\pi}{2} - \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \sin x dx + \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \tan^3 x \left( se c^2 x - 1 \right)dx + \left[ x \right]_{- \frac{\pi}{2}}^\frac{\pi}{2} \]

\[ = \frac{\pi^4}{64} - \frac{\pi^4}{64} + \frac{\pi}{2} - \frac{\pi}{2} - \left[ - \cos x \right]_{- \frac{\pi}{2}}^\frac{\pi}{2} + \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \tan^3 x se c^2 x dx - \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \tan^3 x dx + \frac{\pi}{2} + \frac{\pi}{2}\]

\[ = \pi + 0 + \left[ \frac{\tan^4 x}{4} \right]_{- \frac{\pi}{2}}^\frac{\pi}{2} - \int_{- \frac{\pi}{2}}^\frac{\pi}{2} tanx \sec^2 x dx - \int_{- \frac{\pi}{2}}^\frac{\pi}{2} tan x dx\]

\[ = \pi - \left[ \frac{\tan^2 x}{2} \right]_{- \frac{\pi}{2}}^\frac{\pi}{2} - \left[ - \log\left( \cos x \right) \right]_{- \frac{\pi}{2}}^\frac{\pi}{2} \]

\[ = \pi\]