Calculus Examples

$\frac{4}{\sqrt{x}} + e^{3 x} + \frac{2}{x}$

Step 1

Split the single integral into multiple integrals.

$\int \frac{4}{\sqrt{x}} d x + \int e^{3 x} d x + \int \frac{2}{x} d x$

Step 2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int \frac{1}{\sqrt{x}} d x + \int e^{3 x} d x + \int \frac{2}{x} d x$

Step 3

Apply basic rules of exponents.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$4 \int \frac{1}{x^{\frac{1}{2}}} d x + \int e^{3 x} d x + \int \frac{2}{x} d x$

Step 3.2

Move $x^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 3.3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 3.3.1

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

$4 \int x^{\frac{1}{2} \cdot - 1} d x + \int e^{3 x} d x + \int \frac{2}{x} d x$

Step 3.3.2

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

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

Step 3.3.3

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 5.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 \cdot 1$

Step 5.1.4

Multiply $3$ by $1$ .

$3$

Step 5.2

Rewrite the problem using $u$ and $d u$ .

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

Step 6

Combine $e^{u}$ and $\frac{1}{3}$ .

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

Step 7

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

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

Step 8

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

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

Step 9

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

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

Step 10

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

$4 (2 x^{\frac{1}{2}} + C) + \frac{1}{3} (e^{u} + C) + 2 (\ln (| x |) + C)$

Step 11

Simplify.

$8 x^{\frac{1}{2}} + \frac{e^{u}}{3} + 2 \ln (| x |) + C$

Step 12

Replace all occurrences of $u$ with $3 x$ .

$8 x^{\frac{1}{2}} + \frac{e^{3 x}}{3} + 2 \ln (| x |) + C$

Step 13

Reorder terms.

$8 x^{\frac{1}{2}} + \frac{1}{3} e^{3 x} + 2 \ln (| x |) + C$