How to Integrate by Substitution

The method of integration by substitution for indefinite integrals involves changing a variable say x of an integrand to another variable u. You denote the relationship between x and u as ∫ f (g (x)) g′ (x) dx = ∫ f (u) du. You integrate by substitution when you have a composition of two functions. This method makes the indefinite integral easier to evaluate.

Instructions

1. Use the method that follows to find the solution to the indefinite integral ∫ sin (3x+8) dx.

2. Find inside the integral the composition of two functions and let the u= inner function. In this example the inner function is (3x+8). So set (3x+8) = u.

3. Take the derivative of u= (3x+8) and you find that du= 3 dx. Then rearrange the derivative to get it in the form of du/3= dx.

4. Substitute into the integral in Step 1 everything that depends on x in terms of u. This gives ∫ sin (u) du/3 or ∫1/3 sin (u) du. Now this integral is easier to integrate.

5. Use the Table of Indefinite Integrals and see that ∫c f (u) du= c ∫ f (u) du, c is any constant and ∫sin (u) du= -cos (u) + C. In our case c=1/3.

6. Integrate ∫1/3 sin (u) du with respect to u using the information in Step 5 and you get that 1/3 ∫ sin (u) du = 1/3[-cos (u)] + C.

7. Substitute back in terms of the original variable x and since in Step 2 we set u= (3x+8) you get the final answer of 1/3[-cos (3x+8)] +C where C is a constant.

Tips & Warnings

• To follow this method you need to have a basic understanding of chain rule, derivatives, integration and use of the Table of Indefinite Integrals.