Calculus Examples

$\int_{0}^{π} \int_{0}^{2 π} \cos (θ) \sin (θ) \sin^{2} (x) (\sin (x) - \sin (θ)) d θ d x$

Step 1

Evaluate $\int_{0}^{2 π} \cos (θ) \sin (θ) \sin^{2} (x) (\sin (x) - \sin (θ)) d θ$ .

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Step 1.1

Since $\sin^{2} (x)$ is constant with respect to $θ$ , move $\sin^{2} (x)$ out of the integral.

$\int_{0}^{π} \sin^{2} (x) \int_{0}^{2 π} \cos (θ) \sin (θ) (\sin (x) - \sin (θ)) d θ d x$

Step 1.2

Let $u = \sin (θ)$ . Then $d u = \cos (θ) d θ$ , so $\frac{1}{\cos (θ)} d u = d θ$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.2.1

Let $u = \sin (θ)$ . Find $\frac{d u}{d θ}$ .

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Step 1.2.1.1

Differentiate $\sin (θ)$ .

$\frac{d}{d θ} [\sin (θ)]$

Step 1.2.1.2

The derivative of $\sin (θ)$ with respect to $θ$ is $\cos (θ)$ .

$\cos (θ)$

Step 1.2.2

Substitute the lower limit in for $θ$ in $u = \sin (θ)$ .

$u_{lower} = \sin (0)$

Step 1.2.3

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

$u_{lower} = 0$

Step 1.2.4

Substitute the upper limit in for $θ$ in $u = \sin (θ)$ .

$u_{upper} = \sin (2 π)$

Step 1.2.5

Simplify.

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Step 1.2.5.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$u_{upper} = \sin (0)$

Step 1.2.5.2

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

$u_{upper} = 0$

Step 1.2.6

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

$u_{lower} = 0$

$u_{upper} = 0$

Step 1.2.7

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

$\int_{0}^{π} \sin^{2} (x) \int_{0}^{0} u (\sin (x) - u) d u d x$

Step 1.3

Expand $u (\sin (x) - u)$ .

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Step 1.3.1

Apply the distributive property.

$\int_{0}^{π} \sin^{2} (x) \int_{0}^{0} u \sin (x) + u (- u) d u d x$

Step 1.3.2

Reorder $u$ and $- 1$ .

$\int_{0}^{π} \sin^{2} (x) \int_{0}^{0} u \sin (x) - 1 \cdot u \cdot u d u d x$

Step 1.3.3

Factor out negative.

$\int_{0}^{π} \sin^{2} (x) \int_{0}^{0} u \sin (x) - (u \cdot u) d u d x$

Step 1.3.4

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

$\int_{0}^{π} \sin^{2} (x) \int_{0}^{0} u \sin (x) - (u^{1} u) d u d x$

Step 1.3.5

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

$\int_{0}^{π} \sin^{2} (x) \int_{0}^{0} u \sin (x) - (u^{1} u^{1}) d u d x$

Step 1.3.6

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

$\int_{0}^{π} \sin^{2} (x) \int_{0}^{0} u \sin (x) - u^{1 + 1} d u d x$

Step 1.3.7

Add $1$ and $1$ .

$\int_{0}^{π} \sin^{2} (x) \int_{0}^{0} u \sin (x) - u^{2} d u d x$

Step 1.3.8

Reorder $u \sin (x)$ and $- u^{2}$ .

$\int_{0}^{π} \sin^{2} (x) \int_{0}^{0} - u^{2} + u \sin (x) d u d x$

Step 1.4

Split the single integral into multiple integrals.

$\int_{0}^{π} \sin^{2} (x) (\int_{0}^{0} - u^{2} d u + \int_{0}^{0} u \sin (x) d u) d x$

Step 1.5

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

$\int_{0}^{π} \sin^{2} (x) (- \int_{0}^{0} u^{2} d u + \int_{0}^{0} u \sin (x) d u) d x$

Step 1.6

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

$\int_{0}^{π} \sin^{2} (x) (- (\frac{1}{3} u^{3}]_{0}^{0}) + \int_{0}^{0} u \sin (x) d u) d x$

Step 1.7

Combine $\frac{1}{3}$ and $u^{3}$ .

$\int_{0}^{π} \sin^{2} (x) (- (\frac{u^{3}}{3}]_{0}^{0}) + \int_{0}^{0} u \sin (x) d u) d x$

Step 1.8

Since $\sin (x)$ is constant with respect to $u$ , move $\sin (x)$ out of the integral.

$\int_{0}^{π} \sin^{2} (x) (- (\frac{u^{3}}{3}]_{0}^{0}) + \sin (x) \int_{0}^{0} u d u) d x$

Step 1.9

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

$\int_{0}^{π} \sin^{2} (x) (- (\frac{u^{3}}{3}]_{0}^{0}) + \sin (x) \frac{1}{2} u^{2}]_{0}^{0}) d x$

Step 1.10

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

$\int_{0}^{π} \sin^{2} (x) (- (\frac{u^{3}}{3}]_{0}^{0}) + \sin (x) \frac{u^{2}}{2}]_{0}^{0}) d x$

Step 1.11

Substitute and simplify.

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Step 1.11.1

Evaluate $\frac{u^{3}}{3}$ at $0$ and at $0$ .

$\int_{0}^{π} \sin^{2} (x) (- ((\frac{0^{3}}{3}) - \frac{0^{3}}{3}) + \sin (x) \frac{u^{2}}{2}]_{0}^{0}) d x$

Step 1.11.2

Evaluate $\frac{u^{2}}{2}$ at $0$ and at $0$ .

$\int_{0}^{π} \sin^{2} (x) (- (\frac{0^{3}}{3} - \frac{0^{3}}{3}) + \sin (x) (\frac{0^{2}}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3

Simplify.

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Step 1.11.3.1

Raising $0$ to any positive power yields $0$ .

$\int_{0}^{π} \sin^{2} (x) (- (\frac{0}{3} - \frac{0^{3}}{3}) + \sin (x) (\frac{0^{2}}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.2

Cancel the common factor of $0$ and $3$ .

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Step 1.11.3.2.1

Factor $3$ out of $0$ .

$\int_{0}^{π} \sin^{2} (x) (- (\frac{3 (0)}{3} - \frac{0^{3}}{3}) + \sin (x) (\frac{0^{2}}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.2.2

Cancel the common factors.

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Step 1.11.3.2.2.1

Factor $3$ out of $3$ .

$\int_{0}^{π} \sin^{2} (x) (- (\frac{3 \cdot 0}{3 \cdot 1} - \frac{0^{3}}{3}) + \sin (x) (\frac{0^{2}}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.2.2.2

Cancel the common factor.

$\int_{0}^{π} \sin^{2} (x) (- (\frac{3 \cdot 0}{3 \cdot 1} - \frac{0^{3}}{3}) + \sin (x) (\frac{0^{2}}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.2.2.3

Rewrite the expression.

$\int_{0}^{π} \sin^{2} (x) (- (\frac{0}{1} - \frac{0^{3}}{3}) + \sin (x) (\frac{0^{2}}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.2.2.4

Divide $0$ by $1$ .

$\int_{0}^{π} \sin^{2} (x) (- (0 - \frac{0^{3}}{3}) + \sin (x) (\frac{0^{2}}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.3

Raising $0$ to any positive power yields $0$ .

$\int_{0}^{π} \sin^{2} (x) (- (0 - \frac{0}{3}) + \sin (x) (\frac{0^{2}}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.4

Cancel the common factor of $0$ and $3$ .

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Step 1.11.3.4.1

Factor $3$ out of $0$ .

$\int_{0}^{π} \sin^{2} (x) (- (0 - \frac{3 (0)}{3}) + \sin (x) (\frac{0^{2}}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.4.2

Cancel the common factors.

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Step 1.11.3.4.2.1

Factor $3$ out of $3$ .

$\int_{0}^{π} \sin^{2} (x) (- (0 - \frac{3 \cdot 0}{3 \cdot 1}) + \sin (x) (\frac{0^{2}}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.4.2.2

Cancel the common factor.

$\int_{0}^{π} \sin^{2} (x) (- (0 - \frac{3 \cdot 0}{3 \cdot 1}) + \sin (x) (\frac{0^{2}}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.4.2.3

Rewrite the expression.

$\int_{0}^{π} \sin^{2} (x) (- (0 - \frac{0}{1}) + \sin (x) (\frac{0^{2}}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.4.2.4

Divide $0$ by $1$ .

$\int_{0}^{π} \sin^{2} (x) (- (0 - 0) + \sin (x) (\frac{0^{2}}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.5

Multiply $- 1$ by $0$ .

$\int_{0}^{π} \sin^{2} (x) (- (0 + 0) + \sin (x) (\frac{0^{2}}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.6

Add $0$ and $0$ .

$\int_{0}^{π} \sin^{2} (x) (- 0 + \sin (x) (\frac{0^{2}}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.7

Multiply $- 1$ by $0$ .

$\int_{0}^{π} \sin^{2} (x) (0 + \sin (x) (\frac{0^{2}}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.8

Raising $0$ to any positive power yields $0$ .

$\int_{0}^{π} \sin^{2} (x) (0 + \sin (x) (\frac{0}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.9

Cancel the common factor of $0$ and $2$ .

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Step 1.11.3.9.1

Factor $2$ out of $0$ .

$\int_{0}^{π} \sin^{2} (x) (0 + \sin (x) (\frac{2 (0)}{2} - \frac{0^{2}}{2})) d x$

Step 1.11.3.9.2

Cancel the common factors.

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Step 1.11.3.9.2.1

Factor $2$ out of $2$ .

$\int_{0}^{π} \sin^{2} (x) (0 + \sin (x) (\frac{2 \cdot 0}{2 \cdot 1} - \frac{0^{2}}{2})) d x$

Step 1.11.3.9.2.2

Cancel the common factor.

$\int_{0}^{π} \sin^{2} (x) (0 + \sin (x) (\frac{2 \cdot 0}{2 \cdot 1} - \frac{0^{2}}{2})) d x$

Step 1.11.3.9.2.3

Rewrite the expression.

$\int_{0}^{π} \sin^{2} (x) (0 + \sin (x) (\frac{0}{1} - \frac{0^{2}}{2})) d x$

Step 1.11.3.9.2.4

Divide $0$ by $1$ .

$\int_{0}^{π} \sin^{2} (x) (0 + \sin (x) (0 - \frac{0^{2}}{2})) d x$

Step 1.11.3.10

Raising $0$ to any positive power yields $0$ .

$\int_{0}^{π} \sin^{2} (x) (0 + \sin (x) (0 - \frac{0}{2})) d x$

Step 1.11.3.11

Cancel the common factor of $0$ and $2$ .

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Step 1.11.3.11.1

Factor $2$ out of $0$ .

$\int_{0}^{π} \sin^{2} (x) (0 + \sin (x) (0 - \frac{2 (0)}{2})) d x$

Step 1.11.3.11.2

Cancel the common factors.

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Step 1.11.3.11.2.1

Factor $2$ out of $2$ .

$\int_{0}^{π} \sin^{2} (x) (0 + \sin (x) (0 - \frac{2 \cdot 0}{2 \cdot 1})) d x$

Step 1.11.3.11.2.2

Cancel the common factor.

$\int_{0}^{π} \sin^{2} (x) (0 + \sin (x) (0 - \frac{2 \cdot 0}{2 \cdot 1})) d x$

Step 1.11.3.11.2.3

Rewrite the expression.

$\int_{0}^{π} \sin^{2} (x) (0 + \sin (x) (0 - \frac{0}{1})) d x$

Step 1.11.3.11.2.4

Divide $0$ by $1$ .

$\int_{0}^{π} \sin^{2} (x) (0 + \sin (x) (0 - 0)) d x$

Step 1.11.3.12

Multiply $- 1$ by $0$ .

$\int_{0}^{π} \sin^{2} (x) (0 + \sin (x) (0 + 0)) d x$

Step 1.11.3.13

Add $0$ and $0$ .

$\int_{0}^{π} \sin^{2} (x) (0 + \sin (x) \cdot 0) d x$

Step 1.11.3.14

Multiply $\sin (x)$ by $0$ .

$\int_{0}^{π} \sin^{2} (x) (0 + 0) d x$

Step 1.11.3.15

Add $0$ and $0$ .

$\int_{0}^{π} \sin^{2} (x) \cdot 0 d x$

Step 1.11.3.16

Multiply $\sin^{2} (x)$ by $0$ .

$\int_{0}^{π} 0 d x$

Step 2

Evaluate $\int_{0}^{π} 0 d x$ .

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Step 2.1

The integral of $0$ with respect to $x$ is $0$ .

$0]_{0}^{π}$

Step 2.2

Substitute and simplify.

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Step 2.2.1

Evaluate $0$ at $π$ and at $0$ .

$(0) + 0$

Step 2.2.2

Add $0$ and $0$ .

$0$