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.