Calculus Examples

$\int \frac{10}{\sqrt{2 x + 3}} d x$

Step 1

Let $u = 2 x + 3$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 2 x + 3$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Rewrite.

$\frac{2}{1}$

Step 1.1.2

Divide $2$ by $1$ .

$2$

Step 1.2

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

$\int \frac{5}{\sqrt{u}} d u$

Step 2

Since $5$ is constant with respect to $u$ , move $5$ out of the integral.

$5 \int \frac{1}{\sqrt{u}} d u$

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 $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$5 \int \frac{1}{u^{\frac{1}{2}}} d u$

Step 3.2

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

$5 \int {(u^{\frac{1}{2}})}^{- 1} d u$

Step 3.3

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

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Step 3.3.1

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

$5 \int u^{\frac{1}{2} \cdot - 1} d u$

Step 3.3.2

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

$5 \int u^{\frac{- 1}{2}} d u$

Step 3.3.3

Move the negative in front of the fraction.

$5 \int u^{- \frac{1}{2}} d u$

Step 4

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

$5 (2 u^{\frac{1}{2}} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $5 (2 u^{\frac{1}{2}} + C)$ as $5 \cdot 2 u^{\frac{1}{2}} + C$ .

$5 \cdot 2 u^{\frac{1}{2}} + C$

Step 5.2

Rewrite $5 \cdot 2 u^{\frac{1}{2}} + C$ as $10 u^{\frac{1}{2}} + C$ .

$10 u^{\frac{1}{2}} + C$

Step 6

Replace all occurrences of $u$ with $2 x + 3$ .

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