How to evaluate $\int \frac {\sin x}{7+\cos x^2}dx$
Is there an analytical solution to this?
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4$\begingroup$ It's somewhat unclear if you mean $\cos(x^2)$ or $(\cos x)^2$. If it's the former, then I don't think you'll have any luck finding an analytical solution. If it's the latter, then you would proceed as mathlove did in his answer below. $\endgroup$– Christopher ToniJan 16, 2014 at 12:03
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1 Answer
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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.
