TITLE: Calculating integrals of trigonometric and hyperbolic functions
QUESTION [0 upvotes]: Calculate the following integrals:
A) $\int \frac{x^2}{\sqrt{3-x^2}} \, dx$
B) $\int \frac{2x}{cos^2 (5x)} \, dx$
C) $\int sin(3x+5).sh(x) \, dx$

Well, for A), I tried naming $u=\sqrt{3-x^2}$ but I have problems with $x^2$. I tried calling $u=3-x^2$ but then my problem is $du$ because of the $x$. 
In B), I know that I can take the 2 out, and then call $5x=u$, but what can I do with $\frac{1}{cos^2 (u)}$?
In C) I tried naming $t=3x+5$ so that I got $dt=3dx$, and then I have to compute $\frac{1}{3}\int sin(t).sh(\frac{t-5}{3}) \, dt$, and I thought that I could use integration by parts, but independently on what part I call $u$ and $dv$, the function that I have to integrate is ugly, and also I have to integrate hyperbolic functions, which are "cyclical" when integrating or differentiating.

REPLY [1 votes]: For the B) and C) use integration by parts, the fastest way I used to solve them  is the D.I. Method. There are many videos about it, helps a lot.
$$ \begin{array}{|c|c|c|}
\hline
\text{B)}& D.& I.\\ \hline
 +& x& \sec^2(5x)\\ \hline
 -& 1& \dfrac{1}{5}\tan(5x)\\ \hline
 +& 0& -\dfrac{1}{25}\ln(\cos(5x))\\ \hline
\end{array} \hspace{2.5cm}
\begin{array}{|c|c|c|}
\hline
\text{C)}& D.& I.\\ \hline
 +& \sin(3x+5)& \sinh x\\ \hline
 -& 3\cos(3x+5)& \cosh x\\ \hline
 +& -9\sin(3x+5)& \sinh x\\ \hline
 \end{array}\\$$
$B)\quad 2\int x\sec^2(5x)\ dx= \dfrac{2}{5}x\tan(5x)+\dfrac{1}{25}\ln\left|\cos(5x)\right|+C \\ C)\quad R=\int\sin(3x+5)\sinh x\ dx=\sin(3x+5)\cosh x-3\cos(3x+5)\sinh x-9R\\ \qquad R=\dfrac{1}{10}\Big[\sin(3x+5)\cosh x-3\cos(3x+5)\sinh x\Big]+C \hspace{2cm} \forall\ x\in \mathbb{R} $
Another way to solve C) is using the complex definitions of $\sin x=\frac{e^{-ix}-e^{ix}}{2i}$, (replacing $x$ by $3x+5$) and $\sinh x=\frac{e^{x}-e^{-x}}{2}$.
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