Evaluate $\int \frac {\sin x}{7+\cos x^2} dx$

Question

How to evaluate $\int \frac {\sin x}{7+\cos x^2}dx$

Is there an analytical solution to this?

Answer 1

Suppose that your $\cos x^2$ means $(\cos x)^2=\cos^2x.$

HINT :

Letting $\cos x=u$, you'll get $$\int\frac{-1}{7+u^2}du=\int\frac{-1/7}{1+(u/\sqrt 7)^2}du.$$

In general, if you have $$\int\sin x\cdot f(\cos x)dx,$$ letting $u=\cos x$ would help.

On the other hand, if you have $$\int\cos x\cdot f(\sin x)dx,$$ letting $u=\sin x$ would help.