Calculus Examples

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

Step 1

Let $u = 1 - 2 x^{2}$ . Then $d u = - 4 x d x$ , so $- \frac{1}{4} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 1 - 2 x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $1 - 2 x^{2}$ .

$\frac{d}{d x} [1 - 2 x^{2}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $1 - 2 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 2 x^{2}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- 2 x^{2}]$

Step 1.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 2 x^{2}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 2 x^{2}]$ .

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Step 1.1.3.1

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

$0 - 2 \frac{d}{d x} [x^{2}]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 - 2 (2 x)$

Step 1.1.3.3

Multiply $2$ by $- 2$ .

$0 - 4 x$

Step 1.1.4

Subtract $4 x$ from $0$ .

$- 4 x$

Step 1.2

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

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

Step 2

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Simplify the expression.

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Step 4.1

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

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

Step 4.2

Simplify.

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Step 4.2.1

Apply the product rule to $2 (- u + 1)$ .

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

Step 4.2.2

Move $2^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 4.2.3

Multiply $2$ by $2^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.2.3.1

Multiply $2$ by $2^{- \frac{1}{2}}$ .

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Step 4.2.3.1.1

Raise $2$ to the power of $1$ .

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

Step 4.2.3.1.2

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

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

Step 4.2.3.2

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

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

Step 4.2.3.3

Combine the numerators over the common denominator.

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

Step 4.2.3.4

Subtract $1$ from $2$ .

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

Step 4.3

Apply basic rules of exponents.

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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

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Step 4.3.3.1

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

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

Step 4.3.3.2

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

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

Step 4.3.3.3

Move the negative in front of the fraction.

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

Step 4.3.4

Apply the product rule to $\frac{{(- u + 1)}^{\frac{1}{2}}}{2^{\frac{1}{2}}}$ .

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

Step 4.4

Simplify.

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Step 4.4.1

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

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Step 4.4.1.1

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

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

Step 4.4.1.2

Cancel the common factor of $2$ .

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Step 4.4.1.2.1

Cancel the common factor.

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

Step 4.4.1.2.2

Rewrite the expression.

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

Step 4.4.2

Simplify.

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

Step 4.4.3

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

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Step 4.4.3.1

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

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

Step 4.4.3.2

Cancel the common factor of $2$ .

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Step 4.4.3.2.1

Cancel the common factor.

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

Step 4.4.3.2.2

Rewrite the expression.

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

Step 4.4.4

Evaluate the exponent.

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

Step 4.4.5

Combine $\frac{- u + 1}{2}$ and $u^{- \frac{1}{2}}$ .

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

Step 4.4.6

Move $u^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5

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

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

Step 6

Simplify the expression.

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Step 6.1

Simplify.

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Step 6.1.1

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

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

Step 6.1.2

Multiply $2$ by $4$ .

$- \frac{1}{8} \int \frac{- u + 1}{u^{\frac{1}{2}}} d u$

Step 6.2

Apply basic rules of exponents.

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Step 6.2.1

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

$- \frac{1}{8} \int (- u + 1) {(u^{\frac{1}{2}})}^{- 1} d u$

Step 6.2.2

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

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Step 6.2.2.1

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

$- \frac{1}{8} \int (- u + 1) u^{\frac{1}{2} \cdot - 1} d u$

Step 6.2.2.2

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

$- \frac{1}{8} \int (- u + 1) u^{\frac{- 1}{2}} d u$

Step 6.2.2.3

Move the negative in front of the fraction.

$- \frac{1}{8} \int (- u + 1) u^{- \frac{1}{2}} d u$

Step 7

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

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Step 7.1

Apply the distributive property.

$- \frac{1}{8} \int - u \cdot u^{- \frac{1}{2}} + 1 u^{- \frac{1}{2}} d u$

Step 7.2

Factor out negative.

$- \frac{1}{8} \int - (u \cdot u^{- \frac{1}{2}}) + 1 u^{- \frac{1}{2}} d u$

Step 7.3

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

$- \frac{1}{8} \int - (u^{1} u^{- \frac{1}{2}}) + 1 u^{- \frac{1}{2}} d u$

Step 7.4

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

$- \frac{1}{8} \int - u^{1 - \frac{1}{2}} + 1 u^{- \frac{1}{2}} d u$

Step 7.5

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

$- \frac{1}{8} \int - u^{\frac{2}{2} - \frac{1}{2}} + 1 u^{- \frac{1}{2}} d u$

Step 7.6

Combine the numerators over the common denominator.

$- \frac{1}{8} \int - u^{\frac{2 - 1}{2}} + 1 u^{- \frac{1}{2}} d u$

Step 7.7

Subtract $1$ from $2$ .

$- \frac{1}{8} \int - u^{\frac{1}{2}} + 1 u^{- \frac{1}{2}} d u$

Step 7.8

Multiply $u^{- \frac{1}{2}}$ by $1$ .

$- \frac{1}{8} \int - u^{\frac{1}{2}} + u^{- \frac{1}{2}} d u$

Step 7.9

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

$- \frac{1}{8} \int u^{- \frac{1}{2}} - u^{\frac{1}{2}} d u$

Step 8

Split the single integral into multiple integrals.

$- \frac{1}{8} (\int u^{- \frac{1}{2}} d u + \int - u^{\frac{1}{2}} d u)$

Step 9

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

$- \frac{1}{8} (2 u^{\frac{1}{2}} + C + \int - u^{\frac{1}{2}} d u)$

Step 10

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

$- \frac{1}{8} (2 u^{\frac{1}{2}} + C - \int u^{\frac{1}{2}} d u)$

Step 11

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

$- \frac{1}{8} (2 u^{\frac{1}{2}} + C - (\frac{2}{3} u^{\frac{3}{2}} + C))$

Step 12

Simplify.

$- \frac{1}{8} (2 u^{\frac{1}{2}} - \frac{2}{3} u^{\frac{3}{2}}) + C$

Step 13

Replace all occurrences of $u$ with $1 - 2 x^{2}$ .

$- \frac{1}{8} (2 {(1 - 2 x^{2})}^{\frac{1}{2}} - \frac{2}{3} {(1 - 2 x^{2})}^{\frac{3}{2}}) + C$

Step 14

Simplify.

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Step 14.1

Simplify each term.

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Step 14.1.1

Combine ${(1 - 2 x^{2})}^{\frac{3}{2}}$ and $\frac{2}{3}$ .

$- \frac{1}{8} (2 {(1 - 2 x^{2})}^{\frac{1}{2}} - \frac{{(1 - 2 x^{2})}^{\frac{3}{2}} \cdot 2}{3}) + C$

Step 14.1.2

Move $2$ to the left of ${(1 - 2 x^{2})}^{\frac{3}{2}}$ .

$- \frac{1}{8} (2 {(1 - 2 x^{2})}^{\frac{1}{2}} - \frac{2 {(1 - 2 x^{2})}^{\frac{3}{2}}}{3}) + C$

Step 14.2

To write $2 {(1 - 2 x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{1}{8} (2 {(1 - 2 x^{2})}^{\frac{1}{2}} \cdot \frac{3}{3} - \frac{2 {(1 - 2 x^{2})}^{\frac{3}{2}}}{3}) + C$

Step 14.3

Combine $2 {(1 - 2 x^{2})}^{\frac{1}{2}}$ and $\frac{3}{3}$ .

$- \frac{1}{8} (\frac{2 {(1 - 2 x^{2})}^{\frac{1}{2}} \cdot 3}{3} - \frac{2 {(1 - 2 x^{2})}^{\frac{3}{2}}}{3}) + C$

Step 14.4

Combine the numerators over the common denominator.

$- \frac{1}{8} \cdot \frac{2 {(1 - 2 x^{2})}^{\frac{1}{2}} \cdot 3 - 2 {(1 - 2 x^{2})}^{\frac{3}{2}}}{3} + C$

Step 14.5

Multiply $3$ by $2$ .

$- \frac{1}{8} \cdot \frac{6 {(1 - 2 x^{2})}^{\frac{1}{2}} - 2 {(1 - 2 x^{2})}^{\frac{3}{2}}}{3} + C$

Step 14.6

Multiply $- \frac{1}{8} \cdot \frac{6 {(1 - 2 x^{2})}^{\frac{1}{2}} - 2 {(1 - 2 x^{2})}^{\frac{3}{2}}}{3}$ .

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Step 14.6.1

Multiply $\frac{6 {(1 - 2 x^{2})}^{\frac{1}{2}} - 2 {(1 - 2 x^{2})}^{\frac{3}{2}}}{3}$ by $\frac{1}{8}$ .

$- \frac{6 {(1 - 2 x^{2})}^{\frac{1}{2}} - 2 {(1 - 2 x^{2})}^{\frac{3}{2}}}{3 \cdot 8} + C$

Step 14.6.2

Multiply $3$ by $8$ .

$- \frac{6 {(1 - 2 x^{2})}^{\frac{1}{2}} - 2 {(1 - 2 x^{2})}^{\frac{3}{2}}}{24} + C$

Step 15

Reorder terms.

$- \frac{1}{24} (6 {(1 - 2 x^{2})}^{\frac{1}{2}} - 2 {(1 - 2 x^{2})}^{\frac{3}{2}}) + C$