Calculus Examples

$\int \frac{40 x^{\frac{3}{5}}}{5} - 2 x^{- 3} d x$

Step 1

Simplify.

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Step 1.1

Factor $5$ out of $40 x^{\frac{3}{5}}$ .

$\int \frac{5 (8 x^{\frac{3}{5}})}{5} - 2 x^{- 3} d x$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $5$ out of $5$ .

$\int \frac{5 (8 x^{\frac{3}{5}})}{5 (1)} - 2 x^{- 3} d x$

Step 1.2.2

Cancel the common factor.

$\int \frac{5 (8 x^{\frac{3}{5}})}{5 \cdot 1} - 2 x^{- 3} d x$

Step 1.2.3

Rewrite the expression.

$\int \frac{8 x^{\frac{3}{5}}}{1} - 2 x^{- 3} d x$

Step 1.2.4

Divide $8 x^{\frac{3}{5}}$ by $1$ .

$\int 8 x^{\frac{3}{5}} - 2 x^{- 3} d x$

Step 2

Split the single integral into multiple integrals.

$\int 8 x^{\frac{3}{5}} d x + \int - 2 x^{- 3} d x$

Step 3

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int x^{\frac{3}{5}} d x + \int - 2 x^{- 3} d x$

Step 4

By the Power Rule, the integral of $x^{\frac{3}{5}}$ with respect to $x$ is $\frac{5}{8} x^{\frac{8}{5}}$ .

$8 (\frac{5}{8} x^{\frac{8}{5}} + C) + \int - 2 x^{- 3} d x$

Step 5

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

$8 (\frac{5}{8} x^{\frac{8}{5}} + C) - 2 \int x^{- 3} d x$

Step 6

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

$8 (\frac{5}{8} x^{\frac{8}{5}} + C) - 2 (- \frac{1}{2} x^{- 2} + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

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Step 7.1.1

Combine $\frac{5}{8}$ and $x^{\frac{8}{5}}$ .

$8 (\frac{5 x^{\frac{8}{5}}}{8} + C) - 2 (- \frac{1}{2} x^{- 2} + C)$

Step 7.1.2

Combine $x^{- 2}$ and $\frac{1}{2}$ .

$8 (\frac{5 x^{\frac{8}{5}}}{8} + C) - 2 (- \frac{x^{- 2}}{2} + C)$

Step 7.1.3

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$8 (\frac{5 x^{\frac{8}{5}}}{8} + C) - 2 (- \frac{1}{2 x^{2}} + C)$

Step 7.2

Simplify.

$5 x^{\frac{8}{5}} - 2 (- \frac{1}{2 x^{2}}) + C$

Step 7.3

Simplify.

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Step 7.3.1

Multiply $- 1$ by $- 2$ .

$5 x^{\frac{8}{5}} + 2 \frac{1}{2 x^{2}} + C$

Step 7.3.2

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

$5 x^{\frac{8}{5}} + \frac{2}{2 x^{2}} + C$

Step 7.3.3

Cancel the common factor of $2$ .

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Step 7.3.3.1

Cancel the common factor.

$5 x^{\frac{8}{5}} + \frac{2}{2 x^{2}} + C$

Step 7.3.3.2

Rewrite the expression.

$5 x^{\frac{8}{5}} + \frac{1}{x^{2}} + C$