Question

Solve calculus integral of f(x)=e^3x+30x^2-sin^2x.

Answer 1

The first two terms are pretty easy. For the last term use the identity \(\sin^2 x=\frac{1}{2}(1-\cos(2x))\). So we will then have:
\[\int\limits_{}^{}(e^{3x}+30x^2-{1 \over 2}(1-\cos(2x)))dx={1 \over 3}e^{3x}+10x^3-{1 \over 2}(x-{1 \over 2}\sin(2x))+c\]

Answer 2

Note that:
\[1) \int\limits_{}^{}e^{ax}dx={1 \over a}e^{ax}\]
\[2)\int\limits_{}^{}x^ndx={x^{n+1} \over n+1}\]
\[3)\int\limits_{}^{}\cos(ax)dx={1 \over a}\sin (ax)\]