Question

Solve
$\int {\sin }^{3}xdx+{\cos }^{3}x$

Answer 1

We have,

$\int {\sin }^{3}xdx+{\cos }^{3}x$

Now,

$\int \left(\frac{3\sin x-\sin 3x}{4}\right)dx+{\cos }^{3}x$ $\therefore \sin 3x=3\sin x-4{\sin }^{3}x$

So,

$\int (\frac{3\sin x}{4}-\frac{\sin 3x}{4})dx+{\cos }^{3}x$

$\Rightarrow \int \frac{3\sin x}{4}dx-\int \frac{\sin 3x}{4}dx+{\cos }^{3}x$

$\Rightarrow \frac{3}{4}\int \sin xdx-\frac{1}{4}\int \sin 3xdx+{\cos }^{3}x$

$\Rightarrow \frac{3}{4}(-\cos x)-\frac{1}{4}(-\frac{\cos 3x}{3})+{\cos }^{3}x$

$\Rightarrow -\frac{3}{4}\cos x+\frac{1}{12}\cos 3x+{\cos }^{3}x$

Hence, this is the answer.