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Integrals Involving sin x , cos x and Exponential Functions

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.
All the integrals included in the examples belo are evaluated using Integration by Parts given by:
Integration by Parts Rule
The integration by parts helps in evaluating integral of product of functions of the form U dV/dx .

Examples

In what follows, C is the constant of integration.

Example 1

Evaluate the integral

$\int \sin(x) e^x \, dx$ Solution to Example 1:
Let

$u = \sin(x)$ and

$\dfrac{dv}{dx} = e^x$ which gives

$u' = \cos(x)$ and

$v = \displaystyle \int e^x \, dx = e^x$ .
Use the integration by parts Integration by Parts Rule

as follows

We apply the integration by parts to the term

$\displaystyle \int \cos(x)e^x \, dx$ in the expression above, hence

Simplify the above and rewrite as

Simplify Integral Step 1

Note that the term on the right is the integral we are trying to evaluate, hence the above may be written as follows
Simplify Integral Step 2

Hence the integral is given by
Simplify Integral Step 3

Example 2

Evaluate the integral

$\int \cos(2x) \, e^x \, dx$ Solution to Example 2:
Substitution: Let $u = \cos(2x)$ and $\dfrac{dv}{dx} = e^x$ which gives $u' = - 2 \sin(2x)$ and $v = \displaystyle \int e^x \, dx = e^x$ :
Apply the integration by parts: Integration by Parts Rule
$\displaystyle \int \cos(2x) e^x \, dx = \cos(2x) \, e^x - \int -2 \sin(2x) e^x \, dx$
= $\cos(2x) \, e^x + 2 \int \sin(2x) \, e^x \, dx$
Apply integration by parts to the term on the right
= $\cos(2x) \, e^x + 2 ( \sin(2x)e^x - 2 \displaystyle \int \cos(2x)e^x \, dx )$
= $\cos(2x) \, e^x + 2 \sin(2x) \, e^x - 4 \displaystyle \int \cos(2x)e^x \, dx$
Note that the term on the right is related to the integral we are trying to evaluate, we can write that
$5 \displaystyle \int \cos(2x)e^x \, dx = \cos(2x) \, e^x + 2 \sin(2x) \, e^x$
The given integral is evaluated as $\int \cos(2x) \, e^x \, dx = \dfrac{1}{5} e^x ( \cos(2x) + 2 \sin(2x) ) + C$

Example 3

Evaluate the integral

$\int \sin(3x + 2) \, e^{3x} \, dx$ Solution to Example 3:
Substitution: Let

$u = \sin(3x + 2)$ and

$\dfrac{dv}{dx} = e^{3x}$ which gives

$u' = 3 \cos(3x + 2)$ and

$v = \displaystyle \int e^{3x} \, dx = \dfrac{1}{3} e^{3x}$ .
Apply the integration by parts

$\displaystyle \int \sin(3x + 2) \, e^{3x} \, dx = \sin(3x + 2) \dfrac{1}{3} e^{3x} - \int \cos(3x + 2) \dfrac{1}{3} e^{3x} \, dx$
Apply the integration by parts one more time to the term

$\int \cos(3x + 2) e^{3x} \, dx$

$\displaystyle \int \sin(3x + 2) e^{3x} \, dx = \dfrac{1}{3} \sin(3x + 2) e^{3x} - ( \cos(3x + 2) \dfrac{1}{3} e^{3x} + \int \sin(3x + 2) e^{3x} \, dx )$
Note that the term on the right is the integral to be evaluated, hence the above may be written as

$2 \displaystyle \int \sin(3x + 2) e^{3x} \, dx = \dfrac{1}{3} \sin(3x + 2) e^{3x} = \dfrac{1}{3} \sin(3x + 2) e^{3x} - ( \cos(3x + 2)\dfrac{1}{3} e^{3x}$
Divide all terms by 2 and simplify

$\int \sin(3x + 2) e^{3x} \, dx = \dfrac{1}{6} e^{3x} ( \sin(3x + 2) - \cos(3x + 2) ) + C$

Example 4

Evaluate the integral

$\int \cos(4x) \, e^{2x + 5} \, dx$
Solution to Example 4:
Substitution: Let

$u = \cos(4x)$ and

$\dfrac{dv}{dx} = e^{2x + 5}$ and apply the integration by parts twice

$\displaystyle \int \cos(4x) \, e^{2x + 5} \, dx = \cos(4x) \dfrac{1}{2} e^{2x + 5} + 2 \int \sin(4x) \, e^{2x + 5} \, dx$
=

$\displaystyle \cos(4x) \dfrac{1}{2} e^{2x + 5} + 2 \{ \sin(4x) \dfrac{1}{2}e^{2x + 5} - 2 \int \cos(4x) \, e^{2x + 5} \, dx \}$
The term on the right is the integral to be evaluated, hence

$\int \cos(4x) \, e^{2x + 5} \, dx = \dfrac{1}{10} e^{2x + 5} ( \cos(4x) + 2 \sin(4x) ) + C$

Exercises

Evaluate the following integrals.
1. $\displaystyle\int \cos(x) \, e^x \, dx$
2. $\displaystyle \int \sin(2x) \, e^{3x} \, dx$
3. $\displaystyle \int \cos(-3x + 5) \, e^{5x} \, dx$
4. $\displaystyle \int \sin(-4x + 3) \, e^{-2x + 1} \, dx$

Answers to Above Exercises

1. $\dfrac{1}{2} e^x ( \cos(x) + \sin(x) ) + C$
2. $\dfrac{1}{13} e^{3x} ( 3 \sin(2x) - 2 \cos(2x)) + C$
3. $\dfrac{1}{34} e^{5x} ( 5 \cos(-3x + 5) - 3 \sin(-3x + 5) ) + C$
4. $\dfrac{1}{10} e^{-2x + 1} ( 2 \cos(-4x + 3) - \sin(-4x + 3) ) + C$

Integrals Involving sin x , cos x and Exponential Functions

Examples

Example 1

Example 2

Example 3

Example 4

Exercises

Answers to Above Exercises

More References and links