How to Solve Basic Integration Problems Using Substitution and Partial Fractions

[Krushnaraj Pandya]

Homework Statement

∫(√x)^5/((√x)^7+x^6) dx

Homework Equations

I have learned integration through substitution, trigonometric identities and standard integrals

The Attempt at a Solution

I took (√x)^5 common which gave 1/(x+√x^7), Then I tried to substitute √x=t, √x^7=t but it just seems to complicate things more, I don't see any fruitful trigonometric substitutions either- would be grateful if someone can tell me how to proceed

 

[Mark44]

Homework Statement

∫(√x)^5/((√x)^7+x^6) dx

Homework Equations

I have learned integration through substitution, trigonometric identities and standard integrals

The Attempt at a Solution

I took (√x)^5 common which gave 1/(x+√x^7), Then I tried to substitute √x=t, √x^7=t but it just seems to complicate things more, I don't see any fruitful trigonometric substitutions either- would be grateful if someone can tell me how to proceed

Your substitution should work.
$t = x^{1/2} \Rightarrow t^2 = x \Rightarrow 2tdt = dx$
After you make the substitution, you should get an integral that you can evaluate using partial fractions.

 

[Krushnaraj Pandya]

Your substitution should work.
$t = x^{1/2} \Rightarrow t^2 = x \Rightarrow 2tdt = dx$
After you make the substitution, you should get an integral that you can evaluate using partial fractions.

The exercise in my textbook is supposed to be using the concepts I mentioned under relevant equations, therefore I need to solve it using that

 

[Ray Vickson]

The exercise in my textbook is supposed to be using the concepts I mentioned under relevant equations, therefore I need to solve it using that

Mark44 IS suggesting a method on your list---substitution. After that, partial fractions are a 100% standard method, and anybody who tells you you cannot use them is doing you a dis-service. Some integrals just cannot be done in any other way (except, maybe, by finding them in some "tables of integrals" somewhere).

 

[Krushnaraj Pandya]

Mark44 IS suggesting a method on your list---substitution. After that, partial fractions are a 100% standard method, and anybody who tells you you cannot use them is doing you a dis-service. Some integrals just cannot be done in any other way (except, maybe, by finding them in some "tables of integrals" somewhere).

ohh...ok, I'll use that then and see if I can solve it

 

[Krushnaraj Pandya]

Mark44 IS suggesting a method on your list---substitution. After that, partial fractions are a 100% standard method, and anybody who tells you you cannot use them is doing you a dis-service. Some integrals just cannot be done in any other way (except, maybe, by finding them in some "tables of integrals" somewhere).

I got the correct answer. Thanks a lot!