Calculus Examples

$(5 x + 1) e^{x}$

Step 1

Write $(5 x + 1) e^{x}$ as a function.

$f (x) = (5 x + 1) e^{x}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int (5 x + 1) e^{x} d x$

Step 4

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

$(5 x + 1) e^{x} - \int e^{x} \cdot 5 d x$

Step 5

Move $5$ to the left of $e^{x}$ .

$(5 x + 1) e^{x} - \int 5 e^{x} d x$

Step 6

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

$(5 x + 1) e^{x} - (5 \int e^{x} d x)$

Step 7

Multiply $5$ by $- 1$ .

$(5 x + 1) e^{x} - 5 \int e^{x} d x$

Step 8

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$(5 x + 1) e^{x} - 5 (e^{x} + C)$

Step 9

Simplify the answer.

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Step 9.1

Rewrite $(5 x + 1) e^{x} - 5 (e^{x} + C)$ as $5 x e^{x} + e^{x} - 5 e^{x} + C$ .

$5 x e^{x} + e^{x} - 5 e^{x} + C$

Step 9.2

Subtract $5 e^{x}$ from $e^{x}$ .

$5 x e^{x} - 4 e^{x} + C$

Step 10

The answer is the antiderivative of the function $f (x) = (5 x + 1) e^{x}$ .

$F (x) =$ $5 x e^{x} - 4 e^{x} + C$