₀^{π/4}(2² x+x³+2)dx - NCERT Class XII Mathematics Chapter -

NCERT Class XII Mathematics Chapter - - Solutions

Exercise - 7.9
Evaluate the definite integrals in Exercises 176 to 195.

Question : 192 of 271

Marks: +1, -0

\int\limits_{0}^{\frac{\pi}{4}}(2\sec^2 x+x^3+2)dx

Solution:

Let

I=\int\limits_{0}^{\frac{\pi}{4}}(2\sec^2 x+x^3+2)dx

=\left[2\tan x+\frac{x^4}{4}+2x\right]_{0}^{\frac{\pi}{4}}

=2\left(\tan\frac{\pi}{4}-\tan 0\right)+\frac{1}{4}\left(\frac{\pi^4}{256}-0\right)+2\left(\frac{\pi}{4}-0\right)

=2(1-0)+\frac{\pi^4}{1024}+\frac{\pi}{2}

=\frac{\pi^4}{1024}+\frac{\pi}{2}+2

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