Evaluate integrals: Tutorials with examples and detailed solutions. Also exercises with answers are presented at the end of the page.
Example 1: Evaluate the integral
6 cos x sinx dx
Solution to Example 1:
We first use the trigonometric identity 2sin x cos x = sin (2x) to rewrite the integral as follows
6 cos x sinx dx = 3 sin 2x dx
Substitution: Let u = 2x which leads to du / dx = 2 or du = 2 dx or dx = du / 2, the above integral becomes
6 cos x sinx dx = 3 (1/2) sin u du
We now use integral formulas for sine function to obtain
6 cos x sinx dx = - (3/2) cos u + c
We now substitute u by 2x into the above result to obtain the final result as follows 6 cos x sinx dx = - (3/2) cos 2x + c
As an exercise, differentiate - (3/2) cos 2x + c to obtain 6 sin x cos x which is the integrand in the given integral. This is a way to check the answer to integrals evaluation.
Example 2: Evaluate the integral
x √ (x + 1) dx
Solution to Example 2:
Substitution: Let u = x + 1 which leads to du = dx. We also have x = u - 1. The given integral becomes
x √(x + 1) dx = (u - 1) u^{ 1/2} du = (u^{ 3/2} - u^{ 1/2}) du
We now use property for integral of sum of functions and the formula for integration of power function
= (2 / 5) u^{ 5/2} - (2 / 3) u^{ 3/2} + c
We now substitute u by x + 1 into the above result to obtain the final result as follows
= (2 / 5) (x + 1)^{ 5/2} - (2 / 3) (x + 1)^{ 3/2} + c
+ c
To check the final answer, differentiate the indefinite integral obtained to obtain the integrand x √(x + 1) in the given integral.
Example 3: Evaluate the integral
cos^{ 2} dx
Solution to Example 3:
Use the trigonometric identity cos^{ 2} = (1 + cos(2x)) / 2 to rewrite the given integral as
cos^{ 2} dx = ∫ (1 + cos(2x)) / 2 dx
Substitute: u = 2x so that du = 2 dx and dx = du / 2, and the given integral can be written as
= (1 / 4) (1 + cos(u)) du
Integrate to obtain
= (1 / 4) u + (1 / 4) sin (u) + c
Substitute u by 2x and simplify
= x / 2 + (1 / 4) sin (2x) + c = x / 2 + (1/2) sin x cos x + c
As an exercise, check the final answer by differentiation.
Example 4: Evaluate the integral
x^{ 3} e ^{ x 4} dx
Solution to Example 4:
Substitution: Let u = x ^{ 4} so that du / dx = 4 x^{ 3} which leads to (1 / 4) du = x^{ 3} dx, so that the given integral can be written as
= (1 / 4) e ^{ u} du
We now use formula for integral of exponential function to obtain
= (1 / 4) e ^{ u} + c
Substitute u by u = x ^{ 4}
= (1 / 4) e ^{ x 4} + c
Exercises: Use the table of integrals and the properties above to evaluate the following integrals. [Note that you may need to use more than one of the above properties for one integral].
1. √(x + 1) dx
2. sin ^{ 2} x dx
3. x cos(x^{ 2}) dx
4. x e^{ x 2} dx
Answers to Above Exercises
1. (2 / 3) (x+1)^{ 3/2}
2. x / 2 - (1/2) sin x cos x
3. (1 / 2) sin(x^{ 2})
4. (1 / 2) e^{ x 2}
More references on
integrals and their applications in calculus.