Calculus Examples

$\int \frac{a + b x^{2}}{\sqrt{3 a x + b x^{3}}} d a$

Step 1

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

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Step 1.1

Let $u = 3 a x + b x^{3}$ . Find $\frac{d u}{d a}$ .

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Step 1.1.1

Differentiate $3 a x + b x^{3}$ .

$\frac{d}{d a} [3 a x + b x^{3}]$

Step 1.1.2

By the Sum Rule, the derivative of $3 a x + b x^{3}$ with respect to $a$ is $\frac{d}{d a} [3 a x] + \frac{d}{d a} [b x^{3}]$ .

$\frac{d}{d a} [3 a x] + \frac{d}{d a} [b x^{3}]$

Step 1.1.3

Evaluate $\frac{d}{d a} [3 a x]$ .

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Step 1.1.3.1

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

$3 x \frac{d}{d a} [a] + \frac{d}{d a} [b x^{3}]$

Step 1.1.3.2

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

$3 x \cdot 1 + \frac{d}{d a} [b x^{3}]$

Step 1.1.3.3

Multiply $3$ by $1$ .

$3 x + \frac{d}{d a} [b x^{3}]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $b x^{3}$ is constant with respect to $a$ , the derivative of $b x^{3}$ with respect to $a$ is $0$ .

$3 x + 0$

Step 1.1.4.2

Add $3 x$ and $0$ .

$3 x$

Step 1.2

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

$\int \frac{\frac{u}{3 x} - \frac{b x^{2}}{3} + b x^{2}}{\sqrt{u}} \cdot \frac{1}{3 x} d u$

Step 2

Simplify.

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Step 2.1

To write $b x^{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\int \frac{\frac{u}{3 x} - \frac{b x^{2}}{3} + b x^{2} \cdot \frac{3}{3}}{\sqrt{u}} \cdot \frac{1}{3 x} d u$

Step 2.2

Combine $b x^{2}$ and $\frac{3}{3}$ .

$\int \frac{\frac{u}{3 x} - \frac{b x^{2}}{3} + \frac{b x^{2} \cdot 3}{3}}{\sqrt{u}} \cdot \frac{1}{3 x} d u$

Step 2.3

Combine the numerators over the common denominator.

$\int \frac{\frac{u}{3 x} + \frac{- b x^{2} + b x^{2} \cdot 3}{3}}{\sqrt{u}} \cdot \frac{1}{3 x} d u$

Step 2.4

Move $3$ to the left of $b x^{2}$ .

$\int \frac{\frac{u}{3 x} + \frac{- b x^{2} + 3 \cdot (b x^{2})}{3}}{\sqrt{u}} \cdot \frac{1}{3 x} d u$

Step 2.5

Add $- b x^{2}$ and $3 b x^{2}$ .

$\int \frac{\frac{u}{3 x} + \frac{2 b x^{2}}{3}}{\sqrt{u}} \cdot \frac{1}{3 x} d u$

Step 2.6

Multiply $\frac{\frac{u}{3 x} + \frac{2 b x^{2}}{3}}{\sqrt{u}}$ by $\frac{3 x}{3 x}$ .

$\int \frac{3 x}{3 x} \cdot \frac{\frac{u}{3 x} + \frac{2 b x^{2}}{3}}{\sqrt{u}} \frac{1}{3 x} d u$

Step 2.7

Combine.

$\int \frac{3 x (\frac{u}{3 x} + \frac{2 b x^{2}}{3})}{3 x \sqrt{u}} \cdot \frac{1}{3 x} d u$

Step 2.8

Apply the distributive property.

$\int \frac{3 x \frac{u}{3 x} + 3 x \frac{2 b x^{2}}{3}}{3 x \sqrt{u}} \cdot \frac{1}{3 x} d u$

Step 2.9

Cancel the common factor of $3 x$ .

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Step 2.9.1

Cancel the common factor.

$\int \frac{3 x \frac{u}{3 x} + 3 x \frac{2 b x^{2}}{3}}{3 x \sqrt{u}} \cdot \frac{1}{3 x} d u$

Step 2.9.2

Rewrite the expression.

$\int \frac{u + 3 x \frac{2 b x^{2}}{3}}{3 x \sqrt{u}} \cdot \frac{1}{3 x} d u$

Step 2.10

Cancel the common factor of $3$ .

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Step 2.10.1

Factor $3$ out of $3 x$ .

$\int \frac{u + 3 (x) \frac{2 b x^{2}}{3}}{3 x \sqrt{u}} \cdot \frac{1}{3 x} d u$

Step 2.10.2

Cancel the common factor.

$\int \frac{u + 3 x \frac{2 b x^{2}}{3}}{3 x \sqrt{u}} \cdot \frac{1}{3 x} d u$

Step 2.10.3

Rewrite the expression.

$\int \frac{u + x (2 b x^{2})}{3 x \sqrt{u}} \cdot \frac{1}{3 x} d u$

Step 2.11

Raise $x$ to the power of $1$ .

$\int \frac{u + 2 b (x^{1} x^{2})}{3 x \sqrt{u}} \cdot \frac{1}{3 x} d u$

Step 2.12

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int \frac{u + 2 b x^{1 + 2}}{3 x \sqrt{u}} \cdot \frac{1}{3 x} d u$

Step 2.13

Add $1$ and $2$ .

$\int \frac{u + 2 b x^{3}}{3 x \sqrt{u}} \cdot \frac{1}{3 x} d u$

Step 2.14

Multiply $\frac{u + 2 b x^{3}}{3 x \sqrt{u}}$ by $\frac{1}{3 x}$ .

$\int \frac{u + 2 b x^{3}}{3 x \sqrt{u} (3 x)} d u$

Step 2.15

Multiply $3$ by $3$ .

$\int \frac{u + 2 b x^{3}}{9 x \sqrt{u} x} d u$

Step 2.16

Raise $x$ to the power of $1$ .

$\int \frac{u + 2 b x^{3}}{9 (x^{1} x) \sqrt{u}} d u$

Step 2.17

Raise $x$ to the power of $1$ .

$\int \frac{u + 2 b x^{3}}{9 (x^{1} x^{1}) \sqrt{u}} d u$

Step 2.18

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int \frac{u + 2 b x^{3}}{9 x^{1 + 1} \sqrt{u}} d u$

Step 2.19

Add $1$ and $1$ .

$\int \frac{u + 2 b x^{3}}{9 x^{2} \sqrt{u}} d u$

Step 3

Since $\frac{1}{9 x^{2}}$ is constant with respect to $u$ , move $\frac{1}{9 x^{2}}$ out of the integral.

$\frac{1}{9 x^{2}} \int \frac{u + 2 b x^{3}}{\sqrt{u}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

$\frac{1}{9 x^{2}} \int \frac{u + 2 b x^{3}}{u^{\frac{1}{2}}} d u$

Step 4.2

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

$\frac{1}{9 x^{2}} \int (u + 2 b x^{3}) {(u^{\frac{1}{2}})}^{- 1} d u$

Step 4.3

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

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Step 4.3.1

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

$\frac{1}{9 x^{2}} \int (u + 2 b x^{3}) u^{\frac{1}{2} \cdot - 1} d u$

Step 4.3.2

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

$\frac{1}{9 x^{2}} \int (u + 2 b x^{3}) u^{\frac{- 1}{2}} d u$

Step 4.3.3

Move the negative in front of the fraction.

$\frac{1}{9 x^{2}} \int (u + 2 b x^{3}) u^{- \frac{1}{2}} d u$

Step 5

Expand $(u + 2 b x^{3}) u^{- \frac{1}{2}}$ .

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Step 5.1

Apply the distributive property.

$\frac{1}{9 x^{2}} \int u \cdot u^{- \frac{1}{2}} + 2 b x^{3} u^{- \frac{1}{2}} d u$

Step 5.2

Raise $u$ to the power of $1$ .

$\frac{1}{9 x^{2}} \int u^{1} u^{- \frac{1}{2}} + 2 b x^{3} u^{- \frac{1}{2}} d u$

Step 5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{9 x^{2}} \int u^{1 - \frac{1}{2}} + 2 b x^{3} u^{- \frac{1}{2}} d u$

Step 5.4

Write $1$ as a fraction with a common denominator.

$\frac{1}{9 x^{2}} \int u^{\frac{2}{2} - \frac{1}{2}} + 2 b x^{3} u^{- \frac{1}{2}} d u$

Step 5.5

Combine the numerators over the common denominator.

$\frac{1}{9 x^{2}} \int u^{\frac{2 - 1}{2}} + 2 b x^{3} u^{- \frac{1}{2}} d u$

Step 5.6

Subtract $1$ from $2$ .

$\frac{1}{9 x^{2}} \int u^{\frac{1}{2}} + 2 b x^{3} u^{- \frac{1}{2}} d u$

Step 5.7

Reorder $u^{\frac{1}{2}}$ and $2 b x^{3} u^{- \frac{1}{2}}$ .

$\frac{1}{9 x^{2}} \int 2 b x^{3} u^{- \frac{1}{2}} + u^{\frac{1}{2}} d u$

Step 6

Split the single integral into multiple integrals.

$\frac{1}{9 x^{2}} (\int 2 b x^{3} u^{- \frac{1}{2}} d u + \int u^{\frac{1}{2}} d u)$

Step 7

Since $2 b x^{3}$ is constant with respect to $u$ , move $2 b x^{3}$ out of the integral.

$\frac{1}{9 x^{2}} (2 b x^{3} \int u^{- \frac{1}{2}} d u + \int u^{\frac{1}{2}} d u)$

Step 8

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

$\frac{1}{9 x^{2}} (2 b x^{3} (2 u^{\frac{1}{2}} + C) + \int u^{\frac{1}{2}} d u)$

Step 9

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

$\frac{1}{9 x^{2}} (2 b x^{3} (2 u^{\frac{1}{2}} + C) + \frac{2}{3} u^{\frac{3}{2}} + C)$

Step 10

Simplify.

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Step 10.1

Combine $\frac{2}{3}$ and $u^{\frac{3}{2}}$ .

$\frac{1}{9 x^{2}} (2 b x^{3} (2 u^{\frac{1}{2}} + C) + \frac{2 u^{\frac{3}{2}}}{3} + C)$

Step 10.2

Simplify.

$\frac{1}{9 x^{2}} (4 b x^{3} u^{\frac{1}{2}} + \frac{2 u^{\frac{3}{2}}}{3}) + C$

Step 11

Replace all occurrences of $u$ with $3 a x + b x^{3}$ .

$\frac{1}{9 x^{2}} (4 b x^{3} {(3 a x + b x^{3})}^{\frac{1}{2}} + \frac{2 {(3 a x + b x^{3})}^{\frac{3}{2}}}{3}) + C$

Step 12

Reorder terms.

$\frac{1}{9 x^{2}} (4 b x^{3} {(3 a x + b x^{3})}^{\frac{1}{2}} + \frac{2}{3} {(3 a x + b x^{3})}^{\frac{3}{2}}) + C$