Calculus Examples

$\int_{\sqrt{\frac{π}{2}}}^{\sqrt{π}} 9 θ^{3} \cos (θ^{2}) d θ$

Step 1

Since $9$ is constant with respect to $θ$ , move $9$ out of the integral.

$9 \int_{\sqrt{\frac{π}{2}}}^{\sqrt{π}} θ^{3} \cos (θ^{2}) d θ$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $θ^{2}$ .

$\frac{d}{d θ} [θ^{2}]$

Step 2.1.2

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

$2 θ$

Step 2.2

Substitute the lower limit in for $θ$ in $u = θ^{2}$ .

$u_{lower} = {\sqrt{\frac{π}{2}}}^{2}$

Step 2.3

Simplify.

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Step 2.3.1

Rewrite $\sqrt{\frac{π}{2}}$ as $\frac{\sqrt{π}}{\sqrt{2}}$ .

$u_{lower} = {(\frac{\sqrt{π}}{\sqrt{2}})}^{2}$

Step 2.3.2

Multiply $\frac{\sqrt{π}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$u_{lower} = {(\frac{\sqrt{π}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{2}$

Step 2.3.3

Combine and simplify the denominator.

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Step 2.3.3.1

Multiply $\frac{\sqrt{π}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$u_{lower} = {(\frac{\sqrt{π} \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2}$

Step 2.3.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$u_{lower} = {(\frac{\sqrt{π} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{2}$

Step 2.3.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$u_{lower} = {(\frac{\sqrt{π} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{2}$

Step 2.3.3.4

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

$u_{lower} = {(\frac{\sqrt{π} \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{2}$

Step 2.3.3.5

Add $1$ and $1$ .

$u_{lower} = {(\frac{\sqrt{π} \sqrt{2}}{{\sqrt{2}}^{2}})}^{2}$

Step 2.3.3.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 2.3.3.6.1

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

$u_{lower} = {(\frac{\sqrt{π} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{2}$

Step 2.3.3.6.2

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

$u_{lower} = {(\frac{\sqrt{π} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{2}$

Step 2.3.3.6.3

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

$u_{lower} = {(\frac{\sqrt{π} \sqrt{2}}{2^{\frac{2}{2}}})}^{2}$

Step 2.3.3.6.4

Cancel the common factor of $2$ .

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Step 2.3.3.6.4.1

Cancel the common factor.

$u_{lower} = {(\frac{\sqrt{π} \sqrt{2}}{2^{\frac{2}{2}}})}^{2}$

Step 2.3.3.6.4.2

Rewrite the expression.

$u_{lower} = {(\frac{\sqrt{π} \sqrt{2}}{2^{1}})}^{2}$

Step 2.3.3.6.5

Evaluate the exponent.

$u_{lower} = {(\frac{\sqrt{π} \sqrt{2}}{2})}^{2}$

Step 2.3.4

Combine using the product rule for radicals.

$u_{lower} = {(\frac{\sqrt{π \cdot 2}}{2})}^{2}$

Step 2.3.5

Apply the product rule to $\frac{\sqrt{π \cdot 2}}{2}$ .

$u_{lower} = \frac{{\sqrt{π \cdot 2}}^{2}}{2^{2}}$

Step 2.3.6

Simplify the numerator.

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Step 2.3.6.1

Rewrite ${\sqrt{π \cdot 2}}^{2}$ as $π \cdot 2$ .

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Step 2.3.6.1.1

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

$u_{lower} = \frac{{({(π \cdot 2)}^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 2.3.6.1.2

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

$u_{lower} = \frac{{(π \cdot 2)}^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 2.3.6.1.3

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

$u_{lower} = \frac{{(π \cdot 2)}^{\frac{2}{2}}}{2^{2}}$

Step 2.3.6.1.4

Cancel the common factor of $2$ .

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Step 2.3.6.1.4.1

Cancel the common factor.

$u_{lower} = \frac{{(π \cdot 2)}^{\frac{2}{2}}}{2^{2}}$

Step 2.3.6.1.4.2

Rewrite the expression.

$u_{lower} = \frac{{(π \cdot 2)}^{1}}{2^{2}}$

Step 2.3.6.1.5

Simplify.

$u_{lower} = \frac{π \cdot 2}{2^{2}}$

Step 2.3.6.2

Move $2$ to the left of $π$ .

$u_{lower} = \frac{2 π}{2^{2}}$

Step 2.3.7

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

$u_{lower} = \frac{2 π}{4}$

Step 2.3.8

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

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Step 2.3.8.1

Factor $2$ out of $2 π$ .

$u_{lower} = \frac{2 (π)}{4}$

Step 2.3.8.2

Cancel the common factors.

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Step 2.3.8.2.1

Factor $2$ out of $4$ .

$u_{lower} = \frac{2 π}{2 \cdot 2}$

Step 2.3.8.2.2

Cancel the common factor.

$u_{lower} = \frac{2 π}{2 \cdot 2}$

Step 2.3.8.2.3

Rewrite the expression.

$u_{lower} = \frac{π}{2}$

Step 2.4

Substitute the upper limit in for $θ$ in $u = θ^{2}$ .

$u_{upper} = {\sqrt{π}}^{2}$

Step 2.5

Rewrite ${\sqrt{π}}^{2}$ as $π$ .

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Step 2.5.1

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

$u_{upper} = {(π^{\frac{1}{2}})}^{2}$

Step 2.5.2

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

$u_{upper} = π^{\frac{1}{2} \cdot 2}$

Step 2.5.3

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

$u_{upper} = π^{\frac{2}{2}}$

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

$u_{upper} = π^{\frac{2}{2}}$

Step 2.5.4.2

Rewrite the expression.

$u_{upper} = π^{1}$

Step 2.5.5

Simplify.

$u_{upper} = π$

Step 2.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = \frac{π}{2}$

$u_{upper} = π$

Step 2.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$9 \int_{\frac{π}{2}}^{π} u \cos (u) \frac{1}{2} d u$

Step 3

Simplify.

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Step 3.1

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

$9 \int_{\frac{π}{2}}^{π} \frac{u}{2} \cos (u) d u$

Step 3.2

Combine $\frac{u}{2}$ and $\cos (u)$ .

$9 \int_{\frac{π}{2}}^{π} \frac{u \cos (u)}{2} d u$

Step 4

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

$9 (\frac{1}{2} \int_{\frac{π}{2}}^{π} u \cos (u) d u)$

Step 5

Combine $\frac{1}{2}$ and $9$ .

$\frac{9}{2} \int_{\frac{π}{2}}^{π} u \cos (u) d u$

Step 6

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = u$ and $dv = \cos (u)$ .

$\frac{9}{2} (u \sin (u)]_{\frac{π}{2}}^{π} - \int_{\frac{π}{2}}^{π} \sin (u) d u)$

Step 7

The integral of $\sin (u)$ with respect to $u$ is $- \cos (u)$ .

$\frac{9}{2} (u \sin (u)]_{\frac{π}{2}}^{π} - (- \cos (u)]_{\frac{π}{2}}^{π}))$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $u \sin (u)$ at $π$ and at $\frac{π}{2}$ .

$\frac{9}{2} ((π \sin (π)) - \frac{π}{2} \sin (\frac{π}{2}) - (- \cos (u)]_{\frac{π}{2}}^{π}))$

Step 8.2

Evaluate $- \cos (u)$ at $π$ and at $\frac{π}{2}$ .

$\frac{9}{2} ((π \sin (π)) - \frac{π}{2} \sin (\frac{π}{2}) - ((- \cos (π)) + \cos (\frac{π}{2})))$

Step 8.3

Simplify.

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Step 8.3.1

Combine $\sin (\frac{π}{2})$ and $\frac{π}{2}$ .

$\frac{9}{2} (π \sin (π) - \frac{\sin (\frac{π}{2}) π}{2} - ((- \cos (π)) + \cos (\frac{π}{2})))$

Step 8.3.2

To write $- (- \cos (π) + \cos (\frac{π}{2}))$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{9}{2} (π \sin (π) - \frac{\sin (\frac{π}{2}) π}{2} - (- \cos (π) + \cos (\frac{π}{2})) \cdot \frac{2}{2})$

Step 8.3.3

Combine $- (- \cos (π) + \cos (\frac{π}{2}))$ and $\frac{2}{2}$ .

$\frac{9}{2} (π \sin (π) - \frac{\sin (\frac{π}{2}) π}{2} + \frac{- (- \cos (π) + \cos (\frac{π}{2})) \cdot 2}{2})$

Step 8.3.4

Combine the numerators over the common denominator.

$\frac{9}{2} (π \sin (π) + \frac{- \sin (\frac{π}{2}) π - (- \cos (π) + \cos (\frac{π}{2})) \cdot 2}{2})$

Step 8.3.5

Multiply $2$ by $- 1$ .

$\frac{9}{2} (π \sin (π) + \frac{- \sin (\frac{π}{2}) π - 2 (- \cos (π) + \cos (\frac{π}{2}))}{2})$

Step 9

Simplify.

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Step 9.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{9}{2} (π \sin (π) + \frac{- 1 \cdot 1 π - 2 (- \cos (π) + \cos (\frac{π}{2}))}{2})$

Step 9.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\frac{9}{2} (π \sin (π) + \frac{- 1 \cdot 1 π - 2 (- \cos (π) + 0)}{2})$

Step 9.3

Multiply $- 1$ by $1$ .

$\frac{9}{2} (π \sin (π) + \frac{- π - 2 (- \cos (π) + 0)}{2})$

Step 9.4

Add $- \cos (π)$ and $0$ .

$\frac{9}{2} (π \sin (π) + \frac{- π - 2 (- \cos (π))}{2})$

Step 9.5

Multiply $- 1$ by $- 2$ .

$\frac{9}{2} (π \sin (π) + \frac{- π + 2 \cos (π)}{2})$

Step 9.6

To write $π \sin (π)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{9}{2} (π \sin (π) \cdot \frac{2}{2} + \frac{- π + 2 \cos (π)}{2})$

Step 9.7

Combine $π \sin (π)$ and $\frac{2}{2}$ .

$\frac{9}{2} (\frac{π \sin (π) \cdot 2}{2} + \frac{- π + 2 \cos (π)}{2})$

Step 9.8

Combine the numerators over the common denominator.

$\frac{9}{2} \cdot \frac{π \sin (π) \cdot 2 - π + 2 \cos (π)}{2}$

Step 9.9

Move $2$ to the left of $π \sin (π)$ .

$\frac{9}{2} \cdot \frac{2 \cdot (π \sin (π)) - π + 2 \cos (π)}{2}$

Step 9.10

Multiply $\frac{9}{2}$ by $\frac{2 π \sin (π) - π + 2 \cos (π)}{2}$ .

$\frac{9 (2 π \sin (π) - π + 2 \cos (π))}{2 \cdot 2}$

Step 9.11

Multiply $2$ by $2$ .

$\frac{9 (2 π \sin (π) - π + 2 \cos (π))}{4}$

Step 10

Simplify.

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Step 10.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{9 (2 π \sin (0) - π + 2 \cos (π))}{4}$

Step 10.2

The exact value of $\sin (0)$ is $0$ .

$\frac{9 (2 π \cdot 0 - π + 2 \cos (π))}{4}$

Step 10.3

Multiply $2 π \cdot 0$ .

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Step 10.3.1

Multiply $0$ by $2$ .

$\frac{9 (0 π - π + 2 \cos (π))}{4}$

Step 10.3.2

Multiply $0$ by $π$ .

$\frac{9 (0 - π + 2 \cos (π))}{4}$

Step 10.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{9 (0 - π + 2 (- \cos (0)))}{4}$

Step 10.5

The exact value of $\cos (0)$ is $1$ .

$\frac{9 (0 - π + 2 (- 1 \cdot 1))}{4}$

Step 10.6

Multiply $- 1$ by $1$ .

$\frac{9 (0 - π + 2 \cdot - 1)}{4}$

Step 10.7

Multiply $2$ by $- 1$ .

$\frac{9 (0 - π - 2)}{4}$

Step 10.8

Subtract $π$ from $0$ .

$\frac{9 (- π - 2)}{4}$

Step 10.9

Apply the distributive property.

$\frac{9 (- π) + 9 \cdot - 2}{4}$

Step 10.10

Multiply $- 1$ by $9$ .

$\frac{- 9 π + 9 \cdot - 2}{4}$

Step 10.11

Multiply $9$ by $- 2$ .

$\frac{- 9 π - 18}{4}$

Step 10.12

Factor $- 1$ out of $- 9 π$ .

$\frac{- (9 π) - 18}{4}$

Step 10.13

Rewrite $- 18$ as $- 1 (18)$ .

$\frac{- (9 π) - 1 (18)}{4}$

Step 10.14

Factor $- 1$ out of $- (9 π) - 1 (18)$ .

$\frac{- (9 π + 18)}{4}$

Step 10.15

Rewrite $- (9 π + 18)$ as $- 1 (9 π + 18)$ .

$\frac{- 1 (9 π + 18)}{4}$

Step 10.16

Move the negative in front of the fraction.

$- \frac{9 π + 18}{4}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{9 π + 18}{4}$

Decimal Form:

$- 11.56858347 \dots$