Question

Find the antiderivative $F\left(x\right)$ of the function $f\left(x\right)=\cos 4x$ if $F$ $\left(\frac{\pi }{24}\right)$ = $1$.

Answer 1

$f\left(x\right)=\cos 4x$

$F\left(x\right)=\int f\left(x\right)dx$

$=\int \cos 4xdx$

$=\frac{\sin 4x}{4}+C$

$F\left(\frac{\pi }{24}\right)=\frac{\sin \frac{4\pi }{24}}{4}+C$

$\Rightarrow \frac{\sin \frac{\pi }{6}}{4}+C=1$

$\Rightarrow C=1-\frac{1}{8}$

$\Rightarrow C=\frac{7}{8}$

$F\left(x\right)=\frac{\sin 4x}{4}+\frac{7}{8}$

$=\frac{1}{8}(2\sin 4x+7)$.