Calculus Examples

$\int (\frac{- 4 x - 4 x^{5}}{x^{2}} + 5 e^{7 x}) d x$

Step 1

Remove parentheses.

$\int \frac{- 4 x - 4 x^{5}}{x^{2}} + 5 e^{7 x} d x$

Step 2

Split the single integral into multiple integrals.

$\int \frac{- 4 x - 4 x^{5}}{x^{2}} d x + \int 5 e^{7 x} d x$

Step 3

Apply basic rules of exponents.

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Step 3.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$\int (- 4 x - 4 x^{5}) {(x^{2})}^{- 1} d x + \int 5 e^{7 x} d x$

Step 3.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int (- 4 x - 4 x^{5}) x^{2 \cdot - 1} d x + \int 5 e^{7 x} d x$

Step 3.2.2

Multiply $2$ by $- 1$ .

$\int (- 4 x - 4 x^{5}) x^{- 2} d x + \int 5 e^{7 x} d x$

Step 4

Expand $(- 4 x - 4 x^{5}) x^{- 2}$ .

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Step 4.1

Apply the distributive property.

$\int - 4 x \cdot x^{- 2} - 4 x^{5} x^{- 2} d x + \int 5 e^{7 x} d x$

Step 4.2

Raise $x$ to the power of $1$ .

$\int - 4 (x^{1} x^{- 2}) - 4 x^{5} x^{- 2} d x + \int 5 e^{7 x} d x$

Step 4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int - 4 x^{1 - 2} - 4 x^{5} x^{- 2} d x + \int 5 e^{7 x} d x$

Step 4.4

Subtract $2$ from $1$ .

$\int - 4 x^{- 1} - 4 x^{5} x^{- 2} d x + \int 5 e^{7 x} d x$

Step 4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int - 4 x^{- 1} - 4 x^{5 - 2} d x + \int 5 e^{7 x} d x$

Step 4.6

Subtract $2$ from $5$ .

$\int - 4 x^{- 1} - 4 x^{3} d x + \int 5 e^{7 x} d x$

Step 4.7

Reorder $- 4 x^{- 1}$ and $- 4 x^{3}$ .

$\int - 4 x^{3} - 4 x^{- 1} d x + \int 5 e^{7 x} d x$

Step 5

Split the single integral into multiple integrals.

$\int - 4 x^{3} d x + \int - 4 x^{- 1} d x + \int 5 e^{7 x} d x$

Step 6

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$- 4 \int x^{3} d x + \int - 4 x^{- 1} d x + \int 5 e^{7 x} d x$

Step 7

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$- 4 (\frac{1}{4} x^{4} + C) + \int - 4 x^{- 1} d x + \int 5 e^{7 x} d x$

Step 8

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$- 4 (\frac{1}{4} x^{4} + C) - 4 \int x^{- 1} d x + \int 5 e^{7 x} d x$

Step 9

The integral of $x^{- 1}$ with respect to $x$ is $\ln (| x |)$ .

$- 4 (\frac{1}{4} x^{4} + C) - 4 (\ln (| x |) + C) + \int 5 e^{7 x} d x$

Step 10

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

$- 4 (\frac{1}{4} x^{4} + C) - 4 (\ln (| x |) + C) + 5 \int e^{7 x} d x$

Step 11

Let $u = 7 x$ . Then $d u = 7 d x$ , so $\frac{1}{7} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 11.1

Let $u = 7 x$ . Find $\frac{d u}{d x}$ .

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Step 11.1.1

Differentiate $7 x$ .

$\frac{d}{d x} [7 x]$

Step 11.1.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$7 \frac{d}{d x} [x]$

Step 11.1.3

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

$7 \cdot 1$

Step 11.1.4

Multiply $7$ by $1$ .

$7$

Step 11.2

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

$- 4 (\frac{1}{4} x^{4} + C) - 4 (\ln (| x |) + C) + 5 \int e^{u} \frac{1}{7} d u$

Step 12

Simplify.

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Step 12.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$- 4 (\frac{x^{4}}{4} + C) - 4 (\ln (| x |) + C) + 5 \int e^{u} \frac{1}{7} d u$

Step 12.2

Combine $e^{u}$ and $\frac{1}{7}$ .

$- 4 (\frac{x^{4}}{4} + C) - 4 (\ln (| x |) + C) + 5 \int \frac{e^{u}}{7} d u$

Step 13

Since $\frac{1}{7}$ is constant with respect to $u$ , move $\frac{1}{7}$ out of the integral.

$- 4 (\frac{x^{4}}{4} + C) - 4 (\ln (| x |) + C) + 5 (\frac{1}{7} \int e^{u} d u)$

Step 14

Combine $\frac{1}{7}$ and $5$ .

$- 4 (\frac{x^{4}}{4} + C) - 4 (\ln (| x |) + C) + \frac{5}{7} \int e^{u} d u$

Step 15

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

$- 4 (\frac{x^{4}}{4} + C) - 4 (\ln (| x |) + C) + \frac{5}{7} (e^{u} + C)$

Step 16

Simplify.

$- x^{4} - 4 \ln (| x |) + \frac{5}{7} e^{u} + C$

Step 17

Replace all occurrences of $u$ with $7 x$ .

$- x^{4} - 4 \ln (| x |) + \frac{5}{7} e^{7 x} + C$