Calculus Examples

$\int 100 - \frac{3}{2} \cdot \sqrt{2 q} d q$

Step 1

Split the single integral into multiple integrals.

$\int 100 d q + \int - \frac{3}{2} \cdot \sqrt{2 q} d q$

Step 2

Apply the constant rule.

$100 q + C + \int - \frac{3}{2} \cdot \sqrt{2 q} d q$

Step 3

Since $- \frac{3}{2}$ is constant with respect to $q$ , move $- \frac{3}{2}$ out of the integral.

$100 q + C - \frac{3}{2} \int \sqrt{2 q} d q$

Step 4

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

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Step 4.1

Let $u = 2 q$ . Find $\frac{d u}{d q}$ .

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Step 4.1.1

Differentiate $2 q$ .

$\frac{d}{d q} [2 q]$

Step 4.1.2

Since $2$ is constant with respect to $q$ , the derivative of $2 q$ with respect to $q$ is $2 \frac{d}{d q} [q]$ .

$2 \frac{d}{d q} [q]$

Step 4.1.3

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

$2 \cdot 1$

Step 4.1.4

Multiply $2$ by $1$ .

$2$

Step 4.2

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

$100 q + C - \frac{3}{2} \int \sqrt{u} \frac{1}{2} d u$

Step 5

Combine $\sqrt{u}$ and $\frac{1}{2}$ .

$100 q + C - \frac{3}{2} \int \frac{\sqrt{u}}{2} d u$

Step 6

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$100 q + C - \frac{3}{2} (\frac{1}{2} \int \sqrt{u} d u)$

Step 7

Simplify the expression.

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Step 7.1

Simplify.

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Step 7.1.1

Multiply $\frac{1}{2}$ by $\frac{3}{2}$ .

$100 q + C - \frac{3}{2 \cdot 2} \int \sqrt{u} d u$

Step 7.1.2

Multiply $2$ by $2$ .

$100 q + C - \frac{3}{4} \int \sqrt{u} d u$

Step 7.2

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

$100 q + C - \frac{3}{4} \int u^{\frac{1}{2}} d u$

Step 8

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

$100 q + C - \frac{3}{4} (\frac{2}{3} u^{\frac{3}{2}} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$100 q - \frac{3}{4} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$

Step 9.2

Simplify.

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Step 9.2.1

Multiply $\frac{2}{3}$ by $\frac{3}{4}$ .

$100 q - \frac{2 \cdot 3}{3 \cdot 4} u^{\frac{3}{2}} + C$

Step 9.2.2

Multiply $2$ by $3$ .

$100 q - \frac{6}{3 \cdot 4} u^{\frac{3}{2}} + C$

Step 9.2.3

Multiply $3$ by $4$ .

$100 q - \frac{6}{12} u^{\frac{3}{2}} + C$

Step 9.2.4

Cancel the common factor of $6$ and $12$ .

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Step 9.2.4.1

Factor $6$ out of $6$ .

$100 q - \frac{6 (1)}{12} u^{\frac{3}{2}} + C$

Step 9.2.4.2

Cancel the common factors.

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Step 9.2.4.2.1

Factor $6$ out of $12$ .

$100 q - \frac{6 \cdot 1}{6 \cdot 2} u^{\frac{3}{2}} + C$

Step 9.2.4.2.2

Cancel the common factor.

$100 q - \frac{6 \cdot 1}{6 \cdot 2} u^{\frac{3}{2}} + C$

Step 9.2.4.2.3

Rewrite the expression.

$100 q - \frac{1}{2} u^{\frac{3}{2}} + C$

Step 10

Replace all occurrences of $u$ with $2 q$ .

$100 q - \frac{1}{2} {(2 q)}^{\frac{3}{2}} + C$