Calculus Examples

$\int 3 x^{3} (x^{5} + \frac{4}{x^{4}}) d x$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int 3 x^{3} x^{5} + 3 x^{3} \frac{4}{x^{4}} d x$

Step 1.2

Multiply $x^{3}$ by $x^{5}$ by adding the exponents.

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Step 1.2.1

Move $x^{5}$ .

$\int 3 (x^{5} x^{3}) + 3 x^{3} \frac{4}{x^{4}} d x$

Step 1.2.2

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

$\int 3 x^{5 + 3} + 3 x^{3} \frac{4}{x^{4}} d x$

Step 1.2.3

Add $5$ and $3$ .

$\int 3 x^{8} + 3 x^{3} \frac{4}{x^{4}} d x$

Step 1.3

Cancel the common factor of $x^{3}$ .

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Step 1.3.1

Factor $x^{3}$ out of $3 x^{3}$ .

$\int 3 x^{8} + x^{3} \cdot 3 \frac{4}{x^{4}} d x$

Step 1.3.2

Factor $x^{3}$ out of $x^{4}$ .

$\int 3 x^{8} + x^{3} \cdot 3 \frac{4}{x^{3} x} d x$

Step 1.3.3

Cancel the common factor.

$\int 3 x^{8} + x^{3} \cdot 3 \frac{4}{x^{3} x} d x$

Step 1.3.4

Rewrite the expression.

$\int 3 x^{8} + 3 \frac{4}{x} d x$

Step 1.4

Combine $3$ and $\frac{4}{x}$ .

$\int 3 x^{8} + \frac{3 \cdot 4}{x} d x$

Step 1.5

Multiply $3$ by $4$ .

$\int 3 x^{8} + \frac{12}{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int 3 x^{8} d x + \int \frac{12}{x} d x$

Step 3

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int x^{8} d x + \int \frac{12}{x} d x$

Step 4

By the Power Rule, the integral of $x^{8}$ with respect to $x$ is $\frac{1}{9} x^{9}$ .

$3 (\frac{1}{9} x^{9} + C) + \int \frac{12}{x} d x$

Step 5

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$3 (\frac{1}{9} x^{9} + C) + 12 \int \frac{1}{x} d x$

Step 6

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

$3 (\frac{1}{9} x^{9} + C) + 12 (\ln (| x |) + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

$3 (\frac{1}{9}) x^{9} + 12 \ln (| x |) + C$

Step 7.2

Simplify.

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Step 7.2.1

Combine $3$ and $\frac{1}{9}$ .

$\frac{3}{9} x^{9} + 12 \ln (| x |) + C$

Step 7.2.2

Cancel the common factor of $3$ and $9$ .

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Step 7.2.2.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{9} x^{9} + 12 \ln (| x |) + C$

Step 7.2.2.2

Cancel the common factors.

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Step 7.2.2.2.1

Factor $3$ out of $9$ .

$\frac{3 \cdot 1}{3 \cdot 3} x^{9} + 12 \ln (| x |) + C$

Step 7.2.2.2.2

Cancel the common factor.

$\frac{3 \cdot 1}{3 \cdot 3} x^{9} + 12 \ln (| x |) + C$

Step 7.2.2.2.3

Rewrite the expression.

$\frac{1}{3} x^{9} + 12 \ln (| x |) + C$