Calculus Examples

$\int e^{- \frac{1}{3} x} \cos (5 x) d x$

Step 1

Reorder $e^{- \frac{1}{3} x}$ and $\cos (5 x)$ .

$\int \cos (5 x) e^{- \frac{1}{3} x} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \cos (5 x)$ and $d v = e^{- \frac{1}{3} x}$ .

$\cos (5 x) (- 3 e^{- \frac{1}{3} x}) - \int - 3 e^{- \frac{1}{3} x} (- 5 \sin (5 x)) d x$

Step 3

Since $- 3 \cdot - 5$ is constant with respect to $x$ , move $- 3 \cdot - 5$ out of the integral.

$\cos (5 x) (- 3 e^{- \frac{1}{3} x}) - (- 3 \cdot - 5 \int e^{- \frac{1}{3} x} (\sin (5 x)) d x)$

Step 4

Combine fractions.

Tap for more steps...

Combine $x$ and $\frac{1}{3}$ .

$\cos (5 x) (- 3 e^{- \frac{x}{3}}) - (- 3 \cdot - 5 \int e^{- \frac{1}{3} x} (\sin (5 x)) d x)$

Multiply $- 3$ by $- 5$ .

$\cos (5 x) (- 3 e^{- \frac{x}{3}}) - (15 \int e^{- \frac{1}{3} x} (\sin (5 x)) d x)$

Combine $x$ and $\frac{1}{3}$ .

$\cos (5 x) (- 3 e^{- \frac{x}{3}}) - (15 \int e^{- \frac{x}{3}} \sin (5 x) d x)$

Simplify the expression.

Tap for more steps...

Multiply $15$ by $- 1$ .

$\cos (5 x) (- 3 e^{- \frac{x}{3}}) - 15 \int e^{- \frac{x}{3}} \sin (5 x) d x$

Reorder $e^{- \frac{x}{3}}$ and $\sin (5 x)$ .

$\cos (5 x) (- 3 e^{- \frac{x}{3}}) - 15 \int \sin (5 x) e^{- \frac{x}{3}} d x$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sin (5 x)$ and $d v = e^{- \frac{x}{3}}$ .

$\cos (5 x) (- 3 e^{- \frac{x}{3}}) - 15 (\sin (5 x) (- 3 e^{- \frac{1}{3} x}) - \int - 3 e^{- \frac{1}{3} x} (5 \cos (5 x)) d x)$

Step 6

Since $- 3 \cdot 5$ is constant with respect to $x$ , move $- 3 \cdot 5$ out of the integral.

$\cos (5 x) (- 3 e^{- \frac{x}{3}}) - 15 (\sin (5 x) (- 3 e^{- \frac{1}{3} x}) - (- 3 \cdot 5 \int e^{- \frac{1}{3} x} (\cos (5 x)) d x))$

Step 7

Simplify terms.

Tap for more steps...

Combine $x$ and $\frac{1}{3}$ .

$\cos (5 x) (- 3 e^{- \frac{x}{3}}) - 15 (\sin (5 x) (- 3 e^{- \frac{x}{3}}) - (- 3 \cdot 5 \int e^{- \frac{1}{3} x} (\cos (5 x)) d x))$

Multiply $- 3$ by $5$ .

$\cos (5 x) (- 3 e^{- \frac{x}{3}}) - 15 (\sin (5 x) (- 3 e^{- \frac{x}{3}}) - (- 15 \int e^{- \frac{1}{3} x} (\cos (5 x)) d x))$

Combine $x$ and $\frac{1}{3}$ .

$\cos (5 x) (- 3 e^{- \frac{x}{3}}) - 15 (\sin (5 x) (- 3 e^{- \frac{x}{3}}) - (- 15 \int e^{- \frac{x}{3}} \cos (5 x) d x))$

Multiply $- 15$ by $- 1$ .

$\cos (5 x) (- 3 e^{- \frac{x}{3}}) - 15 (\sin (5 x) (- 3 e^{- \frac{x}{3}}) + 15 \int e^{- \frac{x}{3}} \cos (5 x) d x)$

Apply the distributive property.

$\cos (5 x) (- 3 e^{- \frac{x}{3}}) - 15 (\sin (5 x) (- 3 e^{- \frac{x}{3}})) - 15 (15 \int e^{- \frac{x}{3}} \cos (5 x) d x)$

Multiply.

Tap for more steps...

Multiply $- 3$ by $- 15$ .

$\cos (5 x) (- 3 e^{- \frac{x}{3}}) + 45 (\sin (5 x) (e^{- \frac{x}{3}})) - 15 (15 \int e^{- \frac{x}{3}} \cos (5 x) d x)$

Multiply $15$ by $- 15$ .

$\cos (5 x) (- 3 e^{- \frac{x}{3}}) + 45 (\sin (5 x) (e^{- \frac{x}{3}})) - 225 \int e^{- \frac{x}{3}} \cos (5 x) d x$

Step 8

Solving for $\int e^{- \frac{1}{3} x} \cos (5 x) d x$ , we find that $\int e^{- \frac{1}{3} x} \cos (5 x) d x$ = $\frac{\cos (5 x) (- 3 e^{- \frac{x}{3}}) + 45 (\sin (5 x) (e^{- \frac{x}{3}}))}{226}$ .

$\frac{\cos (5 x) (- 3 e^{- \frac{x}{3}}) + 45 (\sin (5 x) (e^{- \frac{x}{3}}))}{226} + C$

Step 9

Simplify the answer.

Tap for more steps...

Rewrite $\frac{\cos (5 x) (- 3 e^{- \frac{x}{3}}) + 45 \sin (5 x) e^{- \frac{x}{3}}}{226} + C$ as $\frac{- 3 \cos (5 x) e^{- \frac{x}{3}} + 45 \sin (5 x) e^{- \frac{x}{3}}}{226} + C$ .

$\frac{- 3 \cos (5 x) e^{- \frac{x}{3}} + 45 \sin (5 x) e^{- \frac{x}{3}}}{226} + C$

Simplify.

Tap for more steps...

Factor $3 e^{- \frac{x}{3}}$ out of $- 3 \cos (5 x) e^{- \frac{x}{3}} + 45 \sin (5 x) e^{- \frac{x}{3}}$ .

Tap for more steps...

Reorder the expression.

Tap for more steps...

Move $\cos (5 x)$ .

$\frac{- 3 e^{- \frac{x}{3}} \cos (5 x) + 45 \sin (5 x) e^{- \frac{x}{3}}}{226} + C$

Move $\sin (5 x)$ .

$\frac{- 3 e^{- \frac{x}{3}} \cos (5 x) + 45 e^{- \frac{x}{3}} \sin (5 x)}{226} + C$

Factor $3 e^{- \frac{x}{3}}$ out of $- 3 e^{- \frac{x}{3}} \cos (5 x)$ .

$\frac{3 e^{- \frac{x}{3}} (- \cos (5 x)) + 45 e^{- \frac{x}{3}} \sin (5 x)}{226} + C$

Factor $3 e^{- \frac{x}{3}}$ out of $45 e^{- \frac{x}{3}} \sin (5 x)$ .

$\frac{3 e^{- \frac{x}{3}} (- \cos (5 x)) + 3 e^{- \frac{x}{3}} (15 \sin (5 x))}{226} + C$

Factor $3 e^{- \frac{x}{3}}$ out of $3 e^{- \frac{x}{3}} (- \cos (5 x)) + 3 e^{- \frac{x}{3}} (15 \sin (5 x))$ .

$\frac{3 e^{- \frac{x}{3}} (- \cos (5 x) + 15 \sin (5 x))}{226} + C$

Factor $- 1$ out of $- \cos (5 x)$ .

$\frac{3 e^{- \frac{x}{3}} (- (\cos (5 x)) + 15 \sin (5 x))}{226} + C$

Factor $- 1$ out of $15 \sin (5 x)$ .

$\frac{3 e^{- \frac{x}{3}} (- (\cos (5 x)) - (- 15 \sin (5 x)))}{226} + C$

Factor $- 1$ out of $- (\cos (5 x)) - (- 15 \sin (5 x))$ .

$\frac{3 e^{- \frac{x}{3}} (- (\cos (5 x) - 15 \sin (5 x)))}{226} + C$

Rewrite $- (\cos (5 x) - 15 \sin (5 x))$ as $- 1 (\cos (5 x) - 15 \sin (5 x))$ .

$\frac{3 e^{- \frac{x}{3}} (- 1 (\cos (5 x) - 15 \sin (5 x)))}{226} + C$

Move the negative in front of the fraction.

$- \frac{(3 e^{- \frac{x}{3}}) (\cos (5 x) - 15 \sin (5 x))}{226} + C$

Reorder factors in $- \frac{(3 e^{- \frac{x}{3}}) (\cos (5 x) - 15 \sin (5 x))}{226}$ .

$- \frac{3}{226} e^{- \frac{1}{3} x} (\cos (5 x) - 15 \sin (5 x)) + C$