Calculus Examples

$6 \sqrt[3]{x} - \frac{2}{x} + 3 e^{x}$

Step 1

Write $6 \sqrt[3]{x} - \frac{2}{x} + 3 e^{x}$ as a function.

$f (x) = 6 \sqrt[3]{x} - \frac{2}{x} + 3 e^{x}$

Step 2

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 3

Set up the integral to solve.

$F (x) = \int 6 \sqrt[3]{x} - \frac{2}{x} + 3 e^{x} d x$

Step 4

Split the single integral into multiple integrals.

$\int 6 \sqrt[3]{x} d x + \int - \frac{2}{x} d x + \int 3 e^{x} d x$

Step 5

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int \sqrt[3]{x} d x + \int - \frac{2}{x} d x + \int 3 e^{x} d x$

Step 6

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$6 \int x^{\frac{1}{3}} d x + \int - \frac{2}{x} d x + \int 3 e^{x} d x$

Step 7

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

$6 (\frac{3}{4} x^{\frac{4}{3}} + C) + \int - \frac{2}{x} d x + \int 3 e^{x} d x$

Step 8

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$6 (\frac{3}{4} x^{\frac{4}{3}} + C) - \int \frac{2}{x} d x + \int 3 e^{x} d x$

Step 9

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

$6 (\frac{3}{4} x^{\frac{4}{3}} + C) - (2 \int \frac{1}{x} d x) + \int 3 e^{x} d x$

Step 10

Simplify.

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Step 10.1

Combine $\frac{3}{4}$ and $x^{\frac{4}{3}}$ .

$6 (\frac{3 x^{\frac{4}{3}}}{4} + C) - (2 \int \frac{1}{x} d x) + \int 3 e^{x} d x$

Step 10.2

Multiply $2$ by $- 1$ .

$6 (\frac{3 x^{\frac{4}{3}}}{4} + C) - 2 \int \frac{1}{x} d x + \int 3 e^{x} d x$

Step 11

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$6 (\frac{3 x^{\frac{4}{3}}}{4} + C) - 2 (\ln (| x |) + C) + \int 3 e^{x} d x$

Step 12

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

$6 (\frac{3 x^{\frac{4}{3}}}{4} + C) - 2 (\ln (| x |) + C) + 3 \int e^{x} d x$

Step 13

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$6 (\frac{3 x^{\frac{4}{3}}}{4} + C) - 2 (\ln (| x |) + C) + 3 (e^{x} + C)$

Step 14

Simplify.

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Step 14.1

Simplify.

$\frac{9 x^{\frac{4}{3}}}{2} - 2 \ln (| x |) + 3 e^{x} + C$

Step 14.2

Reorder terms.

$\frac{9}{2} x^{\frac{4}{3}} - 2 \ln (| x |) + 3 e^{x} + C$

Step 15

The answer is the antiderivative of the function $f (x) = 6 \sqrt[3]{x} - \frac{2}{x} + 3 e^{x}$ .

$F (x) =$ $\frac{9}{2} x^{\frac{4}{3}} - 2 \ln (| x |) + 3 e^{x} + C$