Indefinite trignometric integral

Question

I tried $u$-substitution and $uv$-substitution, can't seem to figure this out... any help would be appreciated!

Question:

$$\int\frac{x}{\cos(x)}\,dx$$

Thanks!!!

Answer 1

As I already commented $$I=\int\frac{x}{\cos(x)}\,dx$$ looks to be a monster.

What I am ready to bet is that the problem is $$J=\int\frac{x}{\cos^2(x)}\,dx$$ which is a totally different story.

Integrating by parts $u=x$, $v'=\frac{dx}{\cos^2(x)}$, $u'=dx$, $v=\tan(x)$ give $$J=x \tan(x)-\int \tan(x)\,dx=x \tan(x)-\int \frac{\sin(x)}{\cos(x)}\,dx=x \tan(x)+\log(\cos(x))+C$$

Looking deeper $$K_n=\int\frac{x}{\cos^n(x)}\,dx$$ is a monster if $n$ is odd and "quite" simple if $n$ is even.

Answer 2

I tried u-substitution and uv-substitution, can't seem to figure this out...

No wonder you can't, since it is not expressible in terms of elementary functions. See Liouville's theorem and the Risch algorithm for more information. If it were, then the definite integral $\displaystyle\int_0^\tfrac\pi2\dfrac{\tfrac\pi2-x}{\cos x}~dx=2G$ would not require the presence of Catalan's constant in its expression.