Mathematics

# Evaluate the definite integral   $\displaystyle \int_0^{\frac {\pi}{4}}(2\sec^2x+x^3+2)dx$

##### SOLUTION
Let $\displaystyle I=\int_0^{\frac {\pi}{4}}(2\sec^2x+x^3+2)dx$
$\Rightarrow\displaystyle \int (2\sec^2x+x^3+2)dx=2\tan x+\frac {x^4}{4}+2x=F(x)$
By second fundamental theorem of calculus, we obtain
$I=F\left (\frac {\pi}{4}\right )-F(0)$
$\displaystyle =\left \{\left (2\tan \frac {\pi}{4}+\frac {1}{4}\left (\frac {\pi}{4}\right )^4+2\left (\frac {\pi}{4}\right )\right )-(2\tan 0+0+0)\right \}$
$\displaystyle =2 \tan \frac {\pi}{4}+\frac {\pi^4}{4^5}+\frac {\pi}{2}$
$\displaystyle =2+\frac {\pi}{2}+\frac {\pi^4}{1024}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105

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