Can Converting Secant to Cosine Simplify Integration by Parts?

[AStaunton]

problem is to integrate the following by parts:

$$\int x\sec^{2}xdx$$

my feeling is convert the secant term to cosine by:

$$sec^{2}x=cos^{-2}x\Rightarrow\int\sec^{2}xdx=\int\cos^{-2}xdx$$

then:

$$u=\cos^{-2}x\implies du=2\sin x(\cos^{-3}x)$$

and also:

$$dv=xdx\implies v=\frac{x^{2}}{2}$$

however plugging all this into int. by parts equation ends up with:

$$\frac{x^{2}}{2}\cos^{-2}x-\int2\frac{x^{2}}{2}\sin x\cos^{-3}xdx$$

which seems to be an even more complicated integral...

is my idea of converting the secant to cosine a good or does it make more complicated?

any advice appreciated

 

[eumyang]

I wouldn't convert the secant term at all. Remember that sec² x is the derivative of some trig function.

 

[osnarf]

While recognizing the antiderivative of sec^2(x) is the real key to this problem, many times it will make the problem easier if you can eliminate the polynomial portion by choosing that as your u.

 

[SammyS]

While recognizing the antiderivative of sec^2(x) is the real key to this problem, many times it will make the problem easier if you can eliminate the polynomial portion by choosing that as your u.

In this case: both of these ideas will work together.

$$\text{Let: }u=x\quad\text{ and let: }\quad dv=\sec^2(x)\,dx\ .$$

This works out nicely!