Calculus Examples

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

Step 1

Write $e^{- 2 x} + 2 x$ as a function.

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

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.

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

Step 4

Split the single integral into multiple integrals.

$\int e^{- 2 x} d x + \int 2 x d x$

Step 5

Let $u = - 2 x$ . Then $d u = - 2 d x$ , so $- \frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = - 2 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $- 2 x$ .

$\frac{d}{d x} [- 2 x]$

Step 5.1.2

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$- 2 \frac{d}{d x} [x]$

Step 5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 2 \cdot 1$

Step 5.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 5.2

Rewrite the problem using $u$ and $d u$ .

$\int e^{u} \frac{1}{- 2} d u + \int 2 x d x$

Step 6

Simplify.

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Step 6.1

Move the negative in front of the fraction.

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

Step 6.2

Combine $e^{u}$ and $\frac{1}{2}$ .

$\int - \frac{e^{u}}{2} d u + \int 2 x d x$

Step 7

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int \frac{e^{u}}{2} d u + \int 2 x d x$

Step 8

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 9

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

$- \frac{1}{2} (e^{u} + C) + \int 2 x d x$

Step 10

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

$- \frac{1}{2} (e^{u} + C) + 2 \int x d x$

Step 11

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

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

Step 12

Simplify.

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Step 12.1

Simplify.

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

Step 12.2

Simplify.

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Step 12.2.1

Combine $2$ and $\frac{1}{2}$ .

$- \frac{1}{2} e^{u} + \frac{2}{2} x^{2} + C$

Step 12.2.2

Cancel the common factor of $2$ .

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Step 12.2.2.1

Cancel the common factor.

$- \frac{1}{2} e^{u} + \frac{2}{2} x^{2} + C$

Step 12.2.2.2

Rewrite the expression.

$- \frac{1}{2} e^{u} + 1 x^{2} + C$

Step 12.2.3

Multiply $x^{2}$ by $1$ .

$- \frac{1}{2} e^{u} + x^{2} + C$

Step 13

Replace all occurrences of $u$ with $- 2 x$ .

$- \frac{1}{2} e^{- 2 x} + x^{2} + C$

Step 14

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

$F (x) =$ $- \frac{1}{2} e^{- 2 x} + x^{2} + C$