Evaluate integrals: Tutorials with examples and detailed solutions. Also exercises with answers are presented at the end of the page. In what follows, C is called the constant of integration.
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
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.