Calculus Examples

$f (x) = - 3 x^{- 4} + 10 x^{\frac{2}{3}} - 7 x$

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.

$F (x) = \int - 3 x^{- 4} + 10 x^{\frac{2}{3}} - 7 x d x$

Step 3

Split the single integral into multiple integrals.

$\int - 3 x^{- 4} d x + \int 10 x^{\frac{2}{3}} d x + \int - 7 x d x$

Step 4

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

$- 3 \int x^{- 4} d x + \int 10 x^{\frac{2}{3}} d x + \int - 7 x d x$

Step 5

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

$- 3 (- \frac{1}{3} x^{- 3} + C) + \int 10 x^{\frac{2}{3}} d x + \int - 7 x d x$

Step 6

Simplify.

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Step 6.1

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

$- 3 (- \frac{x^{- 3}}{3} + C) + \int 10 x^{\frac{2}{3}} d x + \int - 7 x d x$

Step 6.2

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

$- 3 (- \frac{1}{3 x^{3}} + C) + \int 10 x^{\frac{2}{3}} d x + \int - 7 x d x$

Step 7

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

$- 3 (- \frac{1}{3 x^{3}} + C) + 10 \int x^{\frac{2}{3}} d x + \int - 7 x d x$

Step 8

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

$- 3 (- \frac{1}{3 x^{3}} + C) + 10 (\frac{3}{5} x^{\frac{5}{3}} + C) + \int - 7 x d x$

Step 9

Since $- 7$ is constant with respect to $x$ , move $- 7$ out of the integral.

$- 3 (- \frac{1}{3 x^{3}} + C) + 10 (\frac{3}{5} x^{\frac{5}{3}} + C) - 7 \int x d x$

Step 10

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

$- 3 (- \frac{1}{3 x^{3}} + C) + 10 (\frac{3}{5} x^{\frac{5}{3}} + C) - 7 (\frac{1}{2} x^{2} + C)$

Step 11

Simplify.

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Step 11.1

Simplify.

$\frac{1}{x^{3}} + 6 x^{\frac{5}{3}} - 7 (\frac{1}{2} x^{2}) + C$

Step 11.2

Simplify.

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Step 11.2.1

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

$\frac{1}{x^{3}} + 6 x^{\frac{5}{3}} - 7 \frac{x^{2}}{2} + C$

Step 11.2.2

Combine $- 7$ and $\frac{x^{2}}{2}$ .

$\frac{1}{x^{3}} + 6 x^{\frac{5}{3}} + \frac{- 7 x^{2}}{2} + C$

Step 11.2.3

Move the negative in front of the fraction.

$\frac{1}{x^{3}} + 6 x^{\frac{5}{3}} - \frac{7 x^{2}}{2} + C$

Step 12

Reorder terms.

$\frac{1}{x^{3}} + 6 x^{\frac{5}{3}} - \frac{7}{2} x^{2} + C$

Step 13

The answer is the antiderivative of the function $f (x) = - 3 x^{- 4} + 10 x^{\frac{2}{3}} - 7 x$ .

$F (x) =$ $\frac{1}{x^{3}} + 6 x^{\frac{5}{3}} - \frac{7}{2} x^{2} + C$