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.
\int \sin(x)e^x dx
Let u = sin(x) and dv/dx = e^{x} and then use the integration by parts as follows
\int \sin(x)e^x dx = \sin(x) e^x  \int \cos(x)e^x dx
We apply the integration by parts to the term ∫ cos(x)e^{x} dx in the expression above, hence
\int sin(x) e^x dx = \sin(x) e^x  ( \cos(x)e^x  \int  \sin(x) e^x dx )
Simplify the above and rewrite as
\int sin(x) e^x dx = \sin(x) e^x  \cos(x)e^x  \int \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 \int sin(x) e^x dx = \sin(x) e^x  \cos(x)e^x
Hence the integral is given by
\int sin(x) e^x dx = \dfrac{1}{2} e^x ( \sin(x)  \cos(x))
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}  2∫cos(2x)e^{x} dx } = cos(2x)e^{x} + 2sin(2x)e^{x}  4∫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∫cos(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
Example 3: Evaluate the integral
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^{3x}dx} 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
Example 4: Evaluate the integral
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}  2∫cos(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
Exercises: Evaluate the following integrals.
