How Can I Improve My Integration Techniques?

[pswongaa]

My integration skill is very poor, there a many integral that I can't solve, for example:
$$\int_{0}^{\infty}\frac{\sin^{2n+1} x}{x} dx$$
$$\int_{0}^{\infty}\frac{\cos ax-\cos bx}{x} dx$$
but my friend could solve them very quickly, so may I wonder if there are any books about technique for proper and improper riemann integral? Also what is the best way to master integration?

 

[jedishrfu]

My integration skill is very poor, there a many integral that I can't solve, for example:
$$\int_{0}^{\infty}\frac{\sin^{2n+1} x}{x} dx $$
$$\int_{0}^{\infty}\frac{\cos ax-\cos bx}{x} dx $$
but my friend could solve them very quickly, so may I wonder if there are any books about technique for proper and improper riemann integral? Also what is the best way to master integration?

Something is wrong with your post, the integrals aren't rendering correctly.

Mod note: They're fixed now. They were missing the LaTeX tags and were also malformed. I made corrections, but I'm not 100% certain of what the OP intended.

 

[pswongaa]

Something is wrong with your post, the integrals aren't rendering correctly.

the integrals themselves are not important, I just want to find a way to improve my integration skill

 

[Tsunoyukami]

Unfortunately the only way to get good (ie. fast and accurate) at integrals is by doing integrals - a lot of them.

By going through the procedure repeatedly you will build a collection of tricks that you will learn how to use and when to apply them.

You should open up any calc book and start doing every integral. When you get stuck ask others or look for solutions or
Hints.

 

[SteamKing]

Some integrals can be solved (they have indefinite integrals which are composed of elementary functions) and there are many more which cannot be solved, although the definite integrals can be shown to be equal to a certain value.

It just so happens that the two examples you chose do not have indefinite integrals which are composed of elementary functions.

 

[Crake]

Also what is the best way to master integration?

Integrate! As much as possible.

There's this book that has thousands of exercises in Analysis 1.