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  • Evaluate the definite integrals in Exercises 1 to 20. 17

Evaluate the definite integrals in Exercises 1 to 20.

    Q17.    \int_0^\frac{\pi}{4}(2\sec^2x + x^3 + 2)dx

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Given integral: \int_0^\frac{\pi}{4}(2\sec^2x + x^3 + 2)dx

Consider the integral \int (2\sec^2x + x^3 + 2)dx

\int (2\sec^2x + x^3 + 2)dx = 2\tan x +\frac{x^4}{4}+2x 

So, we have the function of x, f(x) = 2\tan x +\frac{x^4}{4}+2x

Now, by Second fundamental theorem of calculus, we have

I = f(\frac{\pi}{4}) -f(0)

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

=2\tan\frac{\pi}{4} +\frac{\pi^4}{4^5} +\frac{\pi}{2}

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

 

Posted by

Divya Prakash Singh

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