Calculus Examples

$\int \frac{(2 r - 1) \cos (\sqrt{3 {(2 r - 1)}^{2} + 6})}{\sqrt{3 {(2 r - 1)}^{2} + 6}} d r$

Step 1

Let $u_{1} = 3 {(2 r - 1)}^{2} + 6$ . Then $d u_{1} = (24 r - 12) d r$ , so $\frac{1}{12} d u_{1} = (2 r - 1) d r$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = 3 {(2 r - 1)}^{2} + 6$ . Find $\frac{d u_{1}}{d r}$ .

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Step 1.1.1

Differentiate $3 {(2 r - 1)}^{2} + 6$ .

$\frac{d}{d r} [3 {(2 r - 1)}^{2} + 6]$

Step 1.1.2

Rewrite ${(2 r - 1)}^{2}$ as $(2 r - 1) (2 r - 1)$ .

$\frac{d}{d r} [3 ((2 r - 1) (2 r - 1)) + 6]$

Step 1.1.3

Expand $(2 r - 1) (2 r - 1)$ using the FOIL Method.

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Step 1.1.3.1

Apply the distributive property.

$\frac{d}{d r} [3 (2 r (2 r - 1) - 1 (2 r - 1)) + 6]$

Step 1.1.3.2

Apply the distributive property.

$\frac{d}{d r} [3 (2 r (2 r) + 2 r \cdot - 1 - 1 (2 r - 1)) + 6]$

Step 1.1.3.3

Apply the distributive property.

$\frac{d}{d r} [3 (2 r (2 r) + 2 r \cdot - 1 - 1 (2 r) - 1 \cdot - 1) + 6]$

Step 1.1.4

Simplify and combine like terms.

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Step 1.1.4.1

Simplify each term.

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Step 1.1.4.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d r} [3 (2 \cdot 2 r \cdot r + 2 r \cdot - 1 - 1 (2 r) - 1 \cdot - 1) + 6]$

Step 1.1.4.1.2

Multiply $r$ by $r$ by adding the exponents.

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Step 1.1.4.1.2.1

Move $r$ .

$\frac{d}{d r} [3 (2 \cdot 2 (r \cdot r) + 2 r \cdot - 1 - 1 (2 r) - 1 \cdot - 1) + 6]$

Step 1.1.4.1.2.2

Multiply $r$ by $r$ .

$\frac{d}{d r} [3 (2 \cdot 2 r^{2} + 2 r \cdot - 1 - 1 (2 r) - 1 \cdot - 1) + 6]$

Step 1.1.4.1.3

Multiply $2$ by $2$ .

$\frac{d}{d r} [3 (4 r^{2} + 2 r \cdot - 1 - 1 (2 r) - 1 \cdot - 1) + 6]$

Step 1.1.4.1.4

Multiply $- 1$ by $2$ .

$\frac{d}{d r} [3 (4 r^{2} - 2 r - 1 (2 r) - 1 \cdot - 1) + 6]$

Step 1.1.4.1.5

Multiply $2$ by $- 1$ .

$\frac{d}{d r} [3 (4 r^{2} - 2 r - 2 r - 1 \cdot - 1) + 6]$

Step 1.1.4.1.6

Multiply $- 1$ by $- 1$ .

$\frac{d}{d r} [3 (4 r^{2} - 2 r - 2 r + 1) + 6]$

Step 1.1.4.2

Subtract $2 r$ from $- 2 r$ .

$\frac{d}{d r} [3 (4 r^{2} - 4 r + 1) + 6]$

Step 1.1.5

By the Sum Rule, the derivative of $3 (4 r^{2} - 4 r + 1) + 6$ with respect to $r$ is $\frac{d}{d r} [3 (4 r^{2} - 4 r + 1)] + \frac{d}{d r} [6]$ .

$\frac{d}{d r} [3 (4 r^{2} - 4 r + 1)] + \frac{d}{d r} [6]$

Step 1.1.6

Evaluate $\frac{d}{d r} [3 (4 r^{2} - 4 r + 1)]$ .

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Step 1.1.6.1

Since $3$ is constant with respect to $r$ , the derivative of $3 (4 r^{2} - 4 r + 1)$ with respect to $r$ is $3 \frac{d}{d r} [4 r^{2} - 4 r + 1]$ .

$3 \frac{d}{d r} [4 r^{2} - 4 r + 1] + \frac{d}{d r} [6]$

Step 1.1.6.2

By the Sum Rule, the derivative of $4 r^{2} - 4 r + 1$ with respect to $r$ is $\frac{d}{d r} [4 r^{2}] + \frac{d}{d r} [- 4 r] + \frac{d}{d r} [1]$ .

$3 (\frac{d}{d r} [4 r^{2}] + \frac{d}{d r} [- 4 r] + \frac{d}{d r} [1]) + \frac{d}{d r} [6]$

Step 1.1.6.3

Since $4$ is constant with respect to $r$ , the derivative of $4 r^{2}$ with respect to $r$ is $4 \frac{d}{d r} [r^{2}]$ .

$3 (4 \frac{d}{d r} [r^{2}] + \frac{d}{d r} [- 4 r] + \frac{d}{d r} [1]) + \frac{d}{d r} [6]$

Step 1.1.6.4

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

$3 (4 (2 r) + \frac{d}{d r} [- 4 r] + \frac{d}{d r} [1]) + \frac{d}{d r} [6]$

Step 1.1.6.5

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

$3 (4 (2 r) - 4 \frac{d}{d r} [r] + \frac{d}{d r} [1]) + \frac{d}{d r} [6]$

Step 1.1.6.6

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

$3 (4 (2 r) - 4 \cdot 1 + \frac{d}{d r} [1]) + \frac{d}{d r} [6]$

Step 1.1.6.7

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

$3 (4 (2 r) - 4 \cdot 1 + 0) + \frac{d}{d r} [6]$

Step 1.1.6.8

Multiply $2$ by $4$ .

$3 (8 r - 4 \cdot 1 + 0) + \frac{d}{d r} [6]$

Step 1.1.6.9

Multiply $- 4$ by $1$ .

$3 (8 r - 4 + 0) + \frac{d}{d r} [6]$

Step 1.1.6.10

Add $8 r - 4$ and $0$ .

$3 (8 r - 4) + \frac{d}{d r} [6]$

Step 1.1.7

Since $6$ is constant with respect to $r$ , the derivative of $6$ with respect to $r$ is $0$ .

$3 (8 r - 4) + 0$

Step 1.1.8

Simplify.

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Step 1.1.8.1

Apply the distributive property.

$3 (8 r) + 3 \cdot - 4 + 0$

Step 1.1.8.2

Combine terms.

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Step 1.1.8.2.1

Multiply $8$ by $3$ .

$24 r + 3 \cdot - 4 + 0$

Step 1.1.8.2.2

Multiply $3$ by $- 4$ .

$24 r - 12 + 0$

Step 1.1.8.2.3

Add $24 r - 12$ and $0$ .

$24 r - 12$

Step 1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \frac{\cos (\sqrt{u_{1}})}{\sqrt{u_{1}}} \cdot \frac{1}{12} d u_{1}$

Step 2

Simplify.

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Step 2.1

Multiply $\frac{\cos (\sqrt{u_{1}})}{\sqrt{u_{1}}}$ by $\frac{1}{12}$ .

$\int \frac{\cos (\sqrt{u_{1}})}{\sqrt{u_{1}} \cdot 12} d u_{1}$

Step 2.2

Move $12$ to the left of $\sqrt{u_{1}}$ .

$\int \frac{\cos (\sqrt{u_{1}})}{12 \sqrt{u_{1}}} d u_{1}$

Step 3

Since $\frac{1}{12}$ is constant with respect to $u_{1}$ , move $\frac{1}{12}$ out of the integral.

$\frac{1}{12} \int \frac{\cos (\sqrt{u_{1}})}{\sqrt{u_{1}}} d u_{1}$

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

$\frac{1}{12} \int \frac{\cos ({u_{1}}^{\frac{1}{2}})}{\sqrt{u_{1}}} d u_{1}$

Step 4.2

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

$\frac{1}{12} \int \frac{\cos ({u_{1}}^{\frac{1}{2}})}{{u_{1}}^{\frac{1}{2}}} d u_{1}$

Step 4.3

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

$\frac{1}{12} \int \cos ({u_{1}}^{\frac{1}{2}}) {({u_{1}}^{\frac{1}{2}})}^{- 1} d u_{1}$

Step 4.4

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

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Step 4.4.1

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

$\frac{1}{12} \int \cos ({u_{1}}^{\frac{1}{2}}) {u_{1}}^{\frac{1}{2} \cdot - 1} d u_{1}$

Step 4.4.2

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

$\frac{1}{12} \int \cos ({u_{1}}^{\frac{1}{2}}) {u_{1}}^{\frac{- 1}{2}} d u_{1}$

Step 4.4.3

Move the negative in front of the fraction.

$\frac{1}{12} \int \cos ({u_{1}}^{\frac{1}{2}}) {u_{1}}^{- \frac{1}{2}} d u_{1}$

Step 5

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

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Step 5.1

Let $u_{2} = {u_{1}}^{\frac{1}{2}}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 5.1.1

Differentiate ${u_{1}}^{\frac{1}{2}}$ .

$\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}]$

Step 5.1.2

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

$\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1}$

Step 5.1.3

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

$\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 5.1.4

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

$\frac{1}{2} {u_{1}}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}$

Step 5.1.5

Combine the numerators over the common denominator.

$\frac{1}{2} {u_{1}}^{\frac{1 - 1 \cdot 2}{2}}$

Step 5.1.6

Simplify the numerator.

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Step 5.1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {u_{1}}^{\frac{1 - 2}{2}}$

Step 5.1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {u_{1}}^{\frac{- 1}{2}}$

Step 5.1.7

Move the negative in front of the fraction.

$\frac{1}{2} {u_{1}}^{- \frac{1}{2}}$

Step 5.1.8

Simplify.

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Step 5.1.8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} \cdot \frac{1}{{u_{1}}^{\frac{1}{2}}}$

Step 5.1.8.2

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

$\frac{1}{2 {u_{1}}^{\frac{1}{2}}}$

Step 5.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{12} \int 2 \cos (u_{2}) d u_{2}$

Step 6

Since $2$ is constant with respect to $u_{2}$ , move $2$ out of the integral.

$\frac{1}{12} (2 \int \cos (u_{2}) d u_{2})$

Step 7

Simplify.

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Step 7.1

Combine $2$ and $\frac{1}{12}$ .

$\frac{2}{12} \int \cos (u_{2}) d u_{2}$

Step 7.2

Cancel the common factor of $2$ and $12$ .

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Step 7.2.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{12} \int \cos (u_{2}) d u_{2}$

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor $2$ out of $12$ .

$\frac{2 \cdot 1}{2 \cdot 6} \int \cos (u_{2}) d u_{2}$

Step 7.2.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 6} \int \cos (u_{2}) d u_{2}$

Step 7.2.2.3

Rewrite the expression.

$\frac{1}{6} \int \cos (u_{2}) d u_{2}$

Step 8

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{1}{6} (\sin (u_{2}) + C)$

Step 9

Simplify.

$\frac{1}{6} \sin (u_{2}) + C$

Step 10

Substitute back in for each integration substitution variable.

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Step 10.1

Replace all occurrences of $u_{2}$ with ${u_{1}}^{\frac{1}{2}}$ .

$\frac{1}{6} \sin ({u_{1}}^{\frac{1}{2}}) + C$

Step 10.2

Replace all occurrences of $u_{1}$ with $3 {(2 r - 1)}^{2} + 6$ .

$\frac{1}{6} \sin ({(3 {(2 r - 1)}^{2} + 6)}^{\frac{1}{2}}) + C$