Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Factor $x^{4}$ out of $5 x^{7} - 2 x^{4}$ .

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Step 1.1.1

Factor $x^{4}$ out of $5 x^{7}$ .

$\int \frac{x^{4} (5 x^{3}) - 2 x^{4}}{x^{2}} d x$

Step 1.1.2

Factor $x^{4}$ out of $- 2 x^{4}$ .

$\int \frac{x^{4} (5 x^{3}) + x^{4} \cdot - 2}{x^{2}} d x$

Step 1.1.3

Factor $x^{4}$ out of $x^{4} (5 x^{3}) + x^{4} \cdot - 2$ .

$\int \frac{x^{4} (5 x^{3} - 2)}{x^{2}} d x$

Step 1.2

Cancel the common factor of $x^{4}$ and $x^{2}$ .

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Step 1.2.1

Factor $x^{2}$ out of $x^{4} (5 x^{3} - 2)$ .

$\int \frac{x^{2} (x^{2} (5 x^{3} - 2))}{x^{2}} d x$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Multiply by $1$ .

$\int \frac{x^{2} (x^{2} (5 x^{3} - 2))}{x^{2} \cdot 1} d x$

Step 1.2.2.2

Cancel the common factor.

$\int \frac{x^{2} (x^{2} (5 x^{3} - 2))}{x^{2} \cdot 1} d x$

Step 1.2.2.3

Rewrite the expression.

$\int \frac{x^{2} (5 x^{3} - 2)}{1} d x$

Step 1.2.2.4

Divide $x^{2} (5 x^{3} - 2)$ by $1$ .

$\int x^{2} (5 x^{3} - 2) d x$

Step 2

Expand $x^{2} (5 x^{3} - 2)$ .

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Step 2.1

Apply the distributive property.

$\int x^{2} (5 x^{3}) + x^{2} \cdot - 2 d x$

Step 2.2

Reorder $x^{2}$ and $5$ .

$\int 5 \cdot x^{2} x^{3} + x^{2} \cdot - 2 d x$

Step 2.3

Reorder $x^{2}$ and $- 2$ .

$\int 5 \cdot x^{2} x^{3} - 2 \cdot x^{2} d x$

Step 2.4

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

$\int 5 x^{2 + 3} - 2 x^{2} d x$

Step 2.5

Add $2$ and $3$ .

$\int 5 x^{5} - 2 x^{2} d x$

Step 3

Split the single integral into multiple integrals.

$\int 5 x^{5} d x + \int - 2 x^{2} d x$

Step 4

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

$5 \int x^{5} d x + \int - 2 x^{2} d x$

Step 5

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$5 (\frac{1}{6} x^{6} + C) + \int - 2 x^{2} d x$

Step 6

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

$5 (\frac{1}{6} x^{6} + C) - 2 \int x^{2} d x$

Step 7

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$5 (\frac{1}{6} x^{6} + C) - 2 (\frac{1}{3} x^{3} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$5 (\frac{1}{6}) x^{6} - 2 (\frac{1}{3}) x^{3} + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $5$ and $\frac{1}{6}$ .

$\frac{5}{6} x^{6} - 2 (\frac{1}{3}) x^{3} + C$

Step 8.2.2

Combine $- 2$ and $\frac{1}{3}$ .

$\frac{5}{6} x^{6} + \frac{- 2}{3} x^{3} + C$

Step 8.2.3

Move the negative in front of the fraction.

$\frac{5}{6} x^{6} - \frac{2}{3} x^{3} + C$