Calculus Examples

$\int - 2 (5 x + 1) e^{25 x^{2} + 10 x + 25} d x$

Step 1

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

$- 2 \int (5 x + 1) e^{25 x^{2} + 10 x + 25} d x$

Step 2

Let $u = 25 x^{2} + 10 x + 25$ . Then $d u = (50 x + 10) d x$ , so $\frac{1}{10} d u = (5 x + 1) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 25 x^{2} + 10 x + 25$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $25 x^{2} + 10 x + 25$ .

$\frac{d}{d x} [25 x^{2} + 10 x + 25]$

Step 2.1.2

By the Sum Rule, the derivative of $25 x^{2} + 10 x + 25$ with respect to $x$ is $\frac{d}{d x} [25 x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [25]$ .

$\frac{d}{d x} [25 x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [25]$

Step 2.1.3

Evaluate $\frac{d}{d x} [25 x^{2}]$ .

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Step 2.1.3.1

Since $25$ is constant with respect to $x$ , the derivative of $25 x^{2}$ with respect to $x$ is $25 \frac{d}{d x} [x^{2}]$ .

$25 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [25]$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$25 (2 x) + \frac{d}{d x} [10 x] + \frac{d}{d x} [25]$

Step 2.1.3.3

Multiply $2$ by $25$ .

$50 x + \frac{d}{d x} [10 x] + \frac{d}{d x} [25]$

Step 2.1.4

Evaluate $\frac{d}{d x} [10 x]$ .

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Step 2.1.4.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$50 x + 10 \frac{d}{d x} [x] + \frac{d}{d x} [25]$

Step 2.1.4.2

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

$50 x + 10 \cdot 1 + \frac{d}{d x} [25]$

Step 2.1.4.3

Multiply $10$ by $1$ .

$50 x + 10 + \frac{d}{d x} [25]$

Step 2.1.5

Differentiate using the Constant Rule.

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Step 2.1.5.1

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$50 x + 10 + 0$

Step 2.1.5.2

Add $50 x + 10$ and $0$ .

$50 x + 10$

Step 2.2

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

$- 2 \int e^{u} \frac{1}{10} d u$

Step 3

Combine $e^{u}$ and $\frac{1}{10}$ .

$- 2 \int \frac{e^{u}}{10} d u$

Step 4

Since $\frac{1}{10}$ is constant with respect to $u$ , move $\frac{1}{10}$ out of the integral.

$- 2 (\frac{1}{10} \int e^{u} d u)$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{10}$ and $- 2$ .

$\frac{- 2}{10} \int e^{u} d u$

Step 5.2

Cancel the common factor of $- 2$ and $10$ .

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Step 5.2.1

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{10} \int e^{u} d u$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $10$ .

$\frac{2 \cdot - 1}{2 \cdot 5} \int e^{u} d u$

Step 5.2.2.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 \cdot 5} \int e^{u} d u$

Step 5.2.2.3

Rewrite the expression.

$\frac{- 1}{5} \int e^{u} d u$

Step 5.3

Move the negative in front of the fraction.

$- \frac{1}{5} \int e^{u} d u$

Step 6

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

$- \frac{1}{5} (e^{u} + C)$

Step 7

Simplify.

$- \frac{1}{5} e^{u} + C$

Step 8

Replace all occurrences of $u$ with $25 x^{2} + 10 x + 25$ .

$- \frac{1}{5} e^{25 x^{2} + 10 x + 25} + C$