Evaluate Integrals
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 the constant of integration.
ExamplesExample 1Evaluate the integral![]() We first use the trigonometric identity 2sin x cos x = sin (2x) to rewrite the integral as follows ![]() ![]() Substitution: Let u = 2x which leads to du / dx = 2 or du = 2 dx or dx = du / 2, the above integral becomes ![]() ![]() We now use integral formulas for sine function to obtain ![]() We now substitute u by 2x into the above result to obtain the final result as follows ![]() 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 2Evaluate the integral![]() Solution to Example 2: Substitution: Let u = x + 1 which leads to du = dx. We also have x = u - 1. The given integral becomes ![]() ![]() ![]() 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 3Evaluate the integral![]() Solution to Example 3: Use the trigonometric identity cos 2 = (1 + cos(2x)) / 2 to rewrite the given integral as ![]() Substitute: u = 2x so that du = 2 dx and dx = du / 2, and the given integral can be written as = ![]() 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 4Evaluate the integral![]() 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 = ![]() 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
ExercisesUse 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. ![]() 2. ![]() 3. ![]() 4. ![]() Answers to Above Exercises1. (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. |