How to Solve Basic Integration Problems Using Substitution and Partial Fractions

  • Thread starter Krushnaraj Pandya
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In summary, Mark44 is suggesting that substitution can be used to solve an integration exercise. Partial fractions are a standard method for solving integrals that are not found in tables of integrals.
  • #1
Krushnaraj Pandya
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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
 
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  • #2
Krushnaraj Pandya said:

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.
 
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  • #3
Mark44 said:
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
 
  • #4
Krushnaraj Pandya said:
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).
 
  • #5
Ray Vickson said:
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
 
  • #6
Ray Vickson said:
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!
 

Related to How to Solve Basic Integration Problems Using Substitution and Partial Fractions

1. What is basic integration?

Basic integration is a mathematical process used to find the area under a curve by adding up an infinite number of infinitesimally small rectangles. It is also known as finding the antiderivative or the integral of a function.

2. What are the steps to solve a basic integration problem?

The steps to solve a basic integration problem are as follows:

  1. Identify the integrand (the function being integrated).
  2. Apply any necessary algebraic manipulations to simplify the integrand.
  3. Use integration rules to find the antiderivative of the integrand.
  4. Apply any necessary limits of integration and evaluate the antiderivative at these limits.
  5. Simplify the resulting expression to obtain the final answer.

3. What is the difference between indefinite and definite integration?

The main difference between indefinite and definite integration is the presence of limits of integration. Indefinite integration does not have limits of integration, and the result is a general expression with a constant of integration. Definite integration has specific limits of integration, and the result is a numerical value.

4. What are some common integration rules?

Some common integration rules include the power rule, which states that the integral of x^n is (x^(n+1))/(n+1) plus a constant. The constant multiple rule, which states that the integral of k*f(x) is k times the integral of f(x), where k is a constant. And the sum and difference rule, which states that the integral of f(x)±g(x) is the integral of f(x)± the integral of g(x).

5. How can I check my answer for a basic integration problem?

You can check your answer for a basic integration problem by taking the derivative of the result. If the derivative is equal to the original integrand, then your answer is correct. You can also use an online graphing calculator to graph the original function and the antiderivative to visually confirm your answer.

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