Calculus Examples

$f (x) = 2 \sec (x) \tan (x) - 2 x^{3} + \frac{1}{x^{\frac{1}{3}}}$

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.

$F (x) = \int 2 \sec (x) \tan (x) - 2 x^{3} + \frac{1}{x^{\frac{1}{3}}} d x$

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

Since the derivative of $\sec (x)$ is $\sec (x) \tan (x)$ , the integral of $\sec (x) \tan (x)$ is $\sec (x)$ .

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

Step 6

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

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

Step 7

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

$2 (\sec (x) + C) - 2 (\frac{1}{4} x^{4} + C) + \int \frac{1}{x^{\frac{1}{3}}} d x$

Step 8

Simplify the expression.

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Step 8.1

Move $x^{\frac{1}{3}}$ out of the denominator by raising it to the $- 1$ power.

$2 (\sec (x) + C) - 2 (\frac{1}{4} x^{4} + C) + \int {(x^{\frac{1}{3}})}^{- 1} d x$

Step 8.2

Simplify.

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Step 8.2.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$2 (\sec (x) + C) - 2 (\frac{x^{4}}{4} + C) + \int {(x^{\frac{1}{3}})}^{- 1} d x$

Step 8.2.2

Multiply the exponents in ${(x^{\frac{1}{3}})}^{- 1}$ .

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Step 8.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 (\sec (x) + C) - 2 (\frac{x^{4}}{4} + C) + \int x^{\frac{1}{3} \cdot - 1} d x$

Step 8.2.2.2

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

$2 (\sec (x) + C) - 2 (\frac{x^{4}}{4} + C) + \int x^{\frac{- 1}{3}} d x$

Step 8.2.2.3

Move the negative in front of the fraction.

$2 (\sec (x) + C) - 2 (\frac{x^{4}}{4} + C) + \int x^{- \frac{1}{3}} d x$

Step 9

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

$2 (\sec (x) + C) - 2 (\frac{x^{4}}{4} + C) + \frac{3}{2} x^{\frac{2}{3}} + C$

Step 10

Simplify.

$2 \sec (x) - \frac{x^{4}}{2} + \frac{3}{2} x^{\frac{2}{3}} + C$

Step 11

Reorder terms.

$2 \sec (x) - \frac{1}{2} x^{4} + \frac{3}{2} x^{\frac{2}{3}} + C$

Step 12

The answer is the antiderivative of the function $f (x) = 2 \sec (x) \tan (x) - 2 x^{3} + \frac{1}{x^{\frac{1}{3}}}$ .

$F (x) =$ $2 \sec (x) - \frac{1}{2} x^{4} + \frac{3}{2} x^{\frac{2}{3}} + C$