Calculus Examples

$(2 x - 1) e^{x}$

Step 1

Write $(2 x - 1) e^{x}$ as a function.

$f (x) = (2 x - 1) e^{x}$

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.

$F (x) = \int (2 x - 1) e^{x} d x$

Step 4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = 2 x - 1$ and $d v = e^{x}$ .

$(2 x - 1) e^{x} - \int e^{x} \cdot 2 d x$

Step 5

Move $2$ to the left of $e^{x}$ .

$(2 x - 1) e^{x} - \int 2 e^{x} d x$

Step 6

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$(2 x - 1) e^{x} - (2 \int e^{x} d x)$

Step 7

Multiply $2$ by $- 1$ .

$(2 x - 1) e^{x} - 2 \int e^{x} d x$

Step 8

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

$(2 x - 1) e^{x} - 2 (e^{x} + C)$

Step 9

Simplify the answer.

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Step 9.1

Rewrite $(2 x - 1) e^{x} - 2 (e^{x} + C)$ as $2 x e^{x} - e^{x} - 2 e^{x} + C$ .

$2 x e^{x} - e^{x} - 2 e^{x} + C$

Step 9.2

Subtract $2 e^{x}$ from $- e^{x}$ .

$2 x e^{x} - 3 e^{x} + C$

Step 10

The answer is the antiderivative of the function $f (x) = (2 x - 1) e^{x}$ .

$F (x) =$ $2 x e^{x} - 3 e^{x} + C$