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
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