Calculus Examples

$\frac{2}{\sqrt{4 x + 3}} - \frac{4}{x^{5}}$

Step 1

Write $\frac{2}{\sqrt{4 x + 3}} - \frac{4}{x^{5}}$ as a function.

$f (x) = \frac{2}{\sqrt{4 x + 3}} - \frac{4}{x^{5}}$

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.

$F (x) = \int \frac{2}{\sqrt{4 x + 3}} - \frac{4}{x^{5}} d x$

Step 4

Split the single integral into multiple integrals.

$\int \frac{2}{\sqrt{4 x + 3}} d x + \int - \frac{4}{x^{5}} d x$

Step 5

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

$2 \int \frac{1}{\sqrt{4 x + 3}} d x + \int - \frac{4}{x^{5}} d x$

Step 6

Let $u = 4 x + 3$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

Let $u = 4 x + 3$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1

Differentiate $4 x + 3$ .

$\frac{d}{d x} [4 x + 3]$

Step 6.1.2

By the Sum Rule, the derivative of $4 x + 3$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [3]$

Step 6.1.3

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 6.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 6.1.3.2

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

$4 \cdot 1 + \frac{d}{d x} [3]$

Step 6.1.3.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [3]$

Step 6.1.4

Differentiate using the Constant Rule.

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Step 6.1.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$4 + 0$

Step 6.1.4.2

Add $4$ and $0$ .

$4$

Step 6.2

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

$2 \int \frac{1}{\sqrt{u}} \cdot \frac{1}{4} d u + \int - \frac{4}{x^{5}} d x$

Step 7

Simplify.

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Step 7.1

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{4}$ .

$2 \int \frac{1}{\sqrt{u} \cdot 4} d u + \int - \frac{4}{x^{5}} d x$

Step 7.2

Move $4$ to the left of $\sqrt{u}$ .

$2 \int \frac{1}{4 \sqrt{u}} d u + \int - \frac{4}{x^{5}} d x$

Step 8

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$2 (\frac{1}{4} \int \frac{1}{\sqrt{u}} d u) + \int - \frac{4}{x^{5}} d x$

Step 9

Simplify the expression.

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Step 9.1

Simplify.

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Step 9.1.1

Combine $\frac{1}{4}$ and $2$ .

$\frac{2}{4} \int \frac{1}{\sqrt{u}} d u + \int - \frac{4}{x^{5}} d x$

Step 9.1.2

Cancel the common factor of $2$ and $4$ .

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Step 9.1.2.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4} \int \frac{1}{\sqrt{u}} d u + \int - \frac{4}{x^{5}} d x$

Step 9.1.2.2

Cancel the common factors.

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Step 9.1.2.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2} \int \frac{1}{\sqrt{u}} d u + \int - \frac{4}{x^{5}} d x$

Step 9.1.2.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2} \int \frac{1}{\sqrt{u}} d u + \int - \frac{4}{x^{5}} d x$

Step 9.1.2.2.3

Rewrite the expression.

$\frac{1}{2} \int \frac{1}{\sqrt{u}} d u + \int - \frac{4}{x^{5}} d x$

Step 9.2

Apply basic rules of exponents.

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Step 9.2.1

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

$\frac{1}{2} \int \frac{1}{u^{\frac{1}{2}}} d u + \int - \frac{4}{x^{5}} d x$

Step 9.2.2

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

$\frac{1}{2} \int {(u^{\frac{1}{2}})}^{- 1} d u + \int - \frac{4}{x^{5}} d x$

Step 9.2.3

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

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Step 9.2.3.1

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

$\frac{1}{2} \int u^{\frac{1}{2} \cdot - 1} d u + \int - \frac{4}{x^{5}} d x$

Step 9.2.3.2

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

$\frac{1}{2} \int u^{\frac{- 1}{2}} d u + \int - \frac{4}{x^{5}} d x$

Step 9.2.3.3

Move the negative in front of the fraction.

$\frac{1}{2} \int u^{- \frac{1}{2}} d u + \int - \frac{4}{x^{5}} d x$

Step 10

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

$\frac{1}{2} (2 u^{\frac{1}{2}} + C) + \int - \frac{4}{x^{5}} d x$

Step 11

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

$\frac{1}{2} (2 u^{\frac{1}{2}} + C) - \int \frac{4}{x^{5}} d x$

Step 12

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

$\frac{1}{2} (2 u^{\frac{1}{2}} + C) - (4 \int \frac{1}{x^{5}} d x)$

Step 13

Simplify the expression.

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Step 13.1

Multiply $4$ by $- 1$ .

$\frac{1}{2} (2 u^{\frac{1}{2}} + C) - 4 \int \frac{1}{x^{5}} d x$

Step 13.2

Move $x^{5}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{2} (2 u^{\frac{1}{2}} + C) - 4 \int {(x^{5})}^{- 1} d x$

Step 13.3

Multiply the exponents in ${(x^{5})}^{- 1}$ .

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Step 13.3.1

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

$\frac{1}{2} (2 u^{\frac{1}{2}} + C) - 4 \int x^{5 \cdot - 1} d x$

Step 13.3.2

Multiply $5$ by $- 1$ .

$\frac{1}{2} (2 u^{\frac{1}{2}} + C) - 4 \int x^{- 5} d x$

Step 14

By the Power Rule, the integral of $x^{- 5}$ with respect to $x$ is $- \frac{1}{4} x^{- 4}$ .

$\frac{1}{2} (2 u^{\frac{1}{2}} + C) - 4 (- \frac{1}{4} x^{- 4} + C)$

Step 15

Simplify.

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Step 15.1

Simplify.

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Step 15.1.1

Combine $x^{- 4}$ and $\frac{1}{4}$ .

$\frac{1}{2} (2 u^{\frac{1}{2}} + C) - 4 (- \frac{x^{- 4}}{4} + C)$

Step 15.1.2

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} (2 u^{\frac{1}{2}} + C) - 4 (- \frac{1}{4 x^{4}} + C)$

Step 15.2

Simplify.

$u^{\frac{1}{2}} - 4 (- \frac{1}{4 x^{4}}) + C$

Step 15.3

Simplify.

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Step 15.3.1

Multiply $- 1$ by $- 4$ .

$u^{\frac{1}{2}} + 4 \frac{1}{4 x^{4}} + C$

Step 15.3.2

Combine $4$ and $\frac{1}{4 x^{4}}$ .

$u^{\frac{1}{2}} + \frac{4}{4 x^{4}} + C$

Step 15.3.3

Cancel the common factor of $4$ .

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Step 15.3.3.1

Cancel the common factor.

$u^{\frac{1}{2}} + \frac{4}{4 x^{4}} + C$

Step 15.3.3.2

Rewrite the expression.

$u^{\frac{1}{2}} + \frac{1}{x^{4}} + C$

Step 16

Replace all occurrences of $u$ with $4 x + 3$ .

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

Step 17

The answer is the antiderivative of the function $f (x) = \frac{2}{\sqrt{4 x + 3}} - \frac{4}{x^{5}}$ .

$F (x) =$ ${(4 x + 3)}^{\frac{1}{2}} + \frac{1}{x^{4}} + C$