Integral of $2x^2 \sec^2{x} \tan{x}$ I've been trying for a while to find $\int{( 2x^2 \sec^2{x} \tan{x} )} dx$, using integration by parts.
I always end up getting a more complicated integral in the second part of the equation.
For example:
$$ \int{( 2x^2 \sec^2{x} \tan{x} )} dx =
\\ 2x^2 \tan^2x - \int{\tan{x} \cdot \frac{d}{dx}(2x^2 \tan{x})}
\\ \frac{d}{dx}(2x^2 \tan{x})=4x\tan{x} + 2x\sec^2{x} \rightarrow
\\ 2x^2 \tan^2x - \int{4x \tan^2{x}+2x\tan{x}\sec^2{x}}
$$
I've tried integrating with different value for $u$ and $v$, such as:
$$ 1:( 2x^2 \sec^2{x} \tan{x} ),
\\ \tan{x} : 2x^2 \sec^2{x},
\\ 2x^2: \sec^2{x} \tan{x},
\\ \sin{x}: 2x^2 \sec^3{x} $$ etc, however, haven't succeeded.
 A: You can proceed from where you ended up by integrating $4x\tan^2 x$ using the trigonometric identity $\sec^2x = 1 + \tan^2x$ to express it as $4x(\sec^2x - 1)$ first.
Always remember that $\tan^2 x$ is easy to integrate after applying that identity.
A: Use integration by parts with $u=x^2$ and $v=\tan^2(x)$.  Then, we have
$$\int 2x^2 \tan(x)\sec^2(x)\,dx=x^2\tan^2(x)-2\int x\tan^2(x)\,dx$$
Continue with a subsequent integration by parts with $u=x$ and $v=\tan(x)-x$  to obtain
$$\begin{align}
\int 2x^2 \tan(x)\sec^2(x)\,dx&=x^2\tan^2(x)-2\left(x\tan(x)-x^2-\int(\tan(x)-x)\,dx\right)\\\\
&=x^2\tan^2(x)-2x\tan(x)+x^2-2\log(\cos(x))+C\\\\
&=x^2\sec^2(x)-2(x\tan(x)+\log(\cos(x)))+C
\end{align}$$
A: Hint consider whole $\sec^2 (x)\tan (x)=v $ as its integral then apply by parts to get $x^2\tan^2 (x)-(\int 4x\tan^2 (x)) $ again consider $\tan^2 (x) $ as v and apply by parts . Note use $\tan^2 (x)=\sec^2 (x)-1$ and proceed with the integration join both these parts to get the integral
A: We have,
$\int \sec ^{2}x\tan {x}dx$
Put v = $\sec{x}$
Then dv = 2$\sec{x}\tan{x}$
So $\int (v) dv=\frac{1}{2}v^{2}=\frac{1}{2}\sec ^{2}x.$
Now use this result in question,
2$\int{(x^2 \sec^2{x} \tan{x} )} dx$
Integrating by parts, integrate $\sec^2x \tan{x}$ and differentiate $x^2$
We have,
2$\left[\frac{1}{2}x^{2}\sec ^{2}x-\int x\sec
^{2}x\,dx \right]$
Now again integration by parts on $ x\sec
^{2}x\,dx $ by integrating $\sec
^{2}x$ and differentiate x.
You got an answer.
