Tutorial to find integrals involving the product of sin(x) or cos(x) with exponential functions. Exercises with answers are at the bottom of the page.

In what follows, C is the constant of integration.

Example 1: Evaluate the integral

sin(x)e^x dx

Solution to Example 1:

Let u = sin(x) and dv/dx = e^x and then use the integration by parts as follows

sin(x)e^x dx = sin(x)e^x - cos(x)e^x dx

We apply the integration by parts to the term cos(x)e^x dx in the expression above, hence

sin(x)e^x dx = sin(x)e^x - { cos(x)e^x - - sin(x)e^x dx }

Simplify the above and rewrite as

sin(x)e^x dx = sin(x)e^x - cos(x)e^x - sin(x)e^x dx

Note that the term on the right is the integral we are trying to evaluate, hence the above may be written as follows
2 sin(x)e^x dx = sin(x)e^x - cos(x)e^x

Hence the integral is given by

sin(x)e^x dx = (1/2)e^x (sin(x) - cos(x)) + C

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Example 2: Evaluate the integral

cos(2x)e^x dx

Solution to Example 2:

Substitution: Let u = cos(2x) and dv/dx = e^x and apply the integration by parts.

cos(2x)e^x dx = cos(2x)e^x --2sin(2x)e^x dx
= cos(2x)e^x +2sin(2x)e^x dx
Apply integration by parts to the term on the right

= cos(2x)e^x + 2{sin(2x)e^x - 2cos(2x)e^x dx }
= cos(2x)e^x + 2sin(2x)e^x - 4cos(2x)e^x dx
Note that the term on the right is related to the integral we are trying to evaluate, we can write that

5cos(2x)e^x dx = cos(2x)e^x + 2sin(2x)e^x

The given integral is

cos(2x)e^x dx = (1/5)e^x {cos(2x) + 2sin(2x)} + C

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Example 3: Evaluate the integral

sin(3x + 2) e^3x dx

Solution to Example 3:

Substitution: Let u = sin(3x + 2) and dv/dx = e^3x and apply the integration by parts twice.

sin(3x + 2) e^3x dx = sin(3x + 2) (1/3)e^3x -cos(3x + 2)e^3x dx

= (1/3) sin(3x + 2)e^3x - {cos(3x + 2)(1/3)e^3x + sin(3x + 2) e^3xdx}

Note that the term on the right is the integral to be evaluated, hence

sin(3x + 2) e^3x dx = (1/6) e^3x { sin(3x + 2) - cos(3x + 2) } + C

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Example 4: Evaluate the integral

cos(4x) e^{2x + 5} dx

Solution to Example 4:

Substitution: Let u = cos(4x) and dv/dx = e^{2x + 5} and apply the integration by parts twice.

cos(4x) e^{2x + 5} dx = cos(4x) (1/2) e^{2x + 5} + 2 sin(4x) e^{2x + 5} dx

= cos(4x) (1/2) e^{2x + 5} + 2{ sin(4x) (1/2)e^{2x + 5} - 2cos(4x)e^{2x + 5} dx }

The term on the right is the integral to be evaluated, hence

cos(4x) e^{2x + 5} dx = (1/10)e^{2x + 5} {cos(4x) + 2 sin(4x)} + C

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Exercises: Evaluate the following integrals.

1. cos(x) e^x dx

2. sin(2x) e^3x x dx

3. cos(-3x + 5) e^5x) dx

4. sin(-4x + 3) e^{-2x + 1} dx



Answers to Above Exercises

1. (1/2) e^x {cos(x) + sin(x)}

2. (1/13)e^3x {3sin(2x) - 2cos(2x)}

3. (1/34) e^5x { 5cos(-3x + 5) - 3sin(-3x + 5) }

4. (1/10) e^{-2x + 1} { 2cos(-4x + 3) - sin(-4x + 3) }

More references on integrals and their applications in calculus.