Calculus Examples

$f (x) = - 3 x^{- 4} + 10 x^{\frac{2}{3}}$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

$F (x) = \int - 3 x^{- 4} + 10 x^{\frac{2}{3}} d x$

Step 3

Split the single integral into multiple integrals.

$\int - 3 x^{- 4} d x + \int 10 x^{\frac{2}{3}} d x$

Step 4

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

$- 3 \int x^{- 4} d x + \int 10 x^{\frac{2}{3}} d x$

Step 5

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

$- 3 (- \frac{1}{3} x^{- 3} + C) + \int 10 x^{\frac{2}{3}} d x$

Step 6

Simplify.

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Step 6.1

Combine $x^{- 3}$ and $\frac{1}{3}$ .

$- 3 (- \frac{x^{- 3}}{3} + C) + \int 10 x^{\frac{2}{3}} d x$

Step 6.2

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 3 (- \frac{1}{3 x^{3}} + C) + \int 10 x^{\frac{2}{3}} d x$

Step 7

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$- 3 (- \frac{1}{3 x^{3}} + C) + 10 \int x^{\frac{2}{3}} d x$

Step 8

By the Power Rule, the integral of $x^{\frac{2}{3}}$ with respect to $x$ is $\frac{3}{5} x^{\frac{5}{3}}$ .

$- 3 (- \frac{1}{3 x^{3}} + C) + 10 (\frac{3}{5} x^{\frac{5}{3}} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$- \frac{- 1}{x^{3}} + 10 (\frac{3}{5} x^{\frac{5}{3}}) + C$

Step 9.2

Simplify.

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Step 9.2.1

Move the negative in front of the fraction.

$- - \frac{1}{x^{3}} + 10 (\frac{3}{5} x^{\frac{5}{3}}) + C$

Step 9.2.2

Multiply $- 1$ by $- 1$ .

$1 \frac{1}{x^{3}} + 10 (\frac{3}{5} x^{\frac{5}{3}}) + C$

Step 9.2.3

Multiply $\frac{1}{x^{3}}$ by $1$ .

$\frac{1}{x^{3}} + 10 (\frac{3}{5} x^{\frac{5}{3}}) + C$

Step 9.2.4

Combine $\frac{3}{5}$ and $x^{\frac{5}{3}}$ .

$\frac{1}{x^{3}} + 10 \frac{3 x^{\frac{5}{3}}}{5} + C$

Step 9.2.5

Combine $10$ and $\frac{3 x^{\frac{5}{3}}}{5}$ .

$\frac{1}{x^{3}} + \frac{10 (3 x^{\frac{5}{3}})}{5} + C$

Step 9.2.6

Multiply $3$ by $10$ .

$\frac{1}{x^{3}} + \frac{30 x^{\frac{5}{3}}}{5} + C$

Step 9.2.7

Factor $5$ out of $30 x^{\frac{5}{3}}$ .

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

Step 9.2.8

Cancel the common factors.

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Step 9.2.8.1

Factor $5$ out of $5$ .

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

Step 9.2.8.2

Cancel the common factor.

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

Step 9.2.8.3

Rewrite the expression.

$\frac{1}{x^{3}} + \frac{6 x^{\frac{5}{3}}}{1} + C$

Step 9.2.8.4

Divide $6 x^{\frac{5}{3}}$ by $1$ .

$\frac{1}{x^{3}} + 6 x^{\frac{5}{3}} + C$

Step 10

The answer is the antiderivative of the function $f (x) = - 3 x^{- 4} + 10 x^{\frac{2}{3}}$ .

$F (x) =$ $\frac{1}{x^{3}} + 6 x^{\frac{5}{3}} + C$