Calculus Examples

$\int_{0}^{π} {(1 + \cos (7 t))}^{2} \sin (7 t) d t$

Step 1

Let $u_{2} = 1 + \cos (7 t)$ . Then $d u_{2} = - 7 \sin (7 t) d t$ , so $- \frac{1}{7} d u_{2} = \sin (7 t) d t$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.1

Let $u_{2} = 1 + \cos (7 t)$ . Find $\frac{d u_{2}}{d t}$ .

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Step 1.1.1

Differentiate $1 + \cos (7 t)$ .

$\frac{d}{d t} [1 + \cos (7 t)]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $1 + \cos (7 t)$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [\cos (7 t)]$ .

$\frac{d}{d t} [1] + \frac{d}{d t} [\cos (7 t)]$

Step 1.1.2.2

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

$0 + \frac{d}{d t} [\cos (7 t)]$

Step 1.1.3

Evaluate $\frac{d}{d t} [\cos (7 t)]$ .

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Step 1.1.3.1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \cos (t)$ and $g (t) = 7 t$ .

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Step 1.1.3.1.1

To apply the Chain Rule, set $u_{1}$ as $7 t$ .

$0 + \frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d t} [7 t]$

Step 1.1.3.1.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$0 - \sin (u_{1}) \frac{d}{d t} [7 t]$

Step 1.1.3.1.3

Replace all occurrences of $u_{1}$ with $7 t$ .

$0 - \sin (7 t) \frac{d}{d t} [7 t]$

Step 1.1.3.2

Since $7$ is constant with respect to $t$ , the derivative of $7 t$ with respect to $t$ is $7 \frac{d}{d t} [t]$ .

$0 - \sin (7 t) (7 \frac{d}{d t} [t])$

Step 1.1.3.3

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

$0 - \sin (7 t) (7 \cdot 1)$

Step 1.1.3.4

Multiply $7$ by $1$ .

$0 - \sin (7 t) \cdot 7$

Step 1.1.3.5

Multiply $7$ by $- 1$ .

$0 - 7 \sin (7 t)$

Step 1.1.4

Subtract $7 \sin (7 t)$ from $0$ .

$- 7 \sin (7 t)$

Step 1.2

Substitute the lower limit in for $t$ in $u_{2} = 1 + \cos (7 t)$ .

$u_{lower} = 1 + \cos (7 \cdot 0)$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $7$ by $0$ .

$u_{lower} = 1 + \cos (0)$

Step 1.3.1.2

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

$u_{lower} = 1 + 1$

Step 1.3.2

Add $1$ and $1$ .

$u_{lower} = 2$

Step 1.4

Substitute the upper limit in for $t$ in $u_{2} = 1 + \cos (7 t)$ .

$u_{upper} = 1 + \cos (7 π)$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

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

$u_{upper} = 1 + \cos (π)$

Step 1.5.1.2

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.

$u_{upper} = 1 - \cos (0)$

Step 1.5.1.3

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

$u_{upper} = 1 - 1 \cdot 1$

Step 1.5.1.4

Multiply $- 1$ by $1$ .

$u_{upper} = 1 - 1$

Step 1.5.2

Subtract $1$ from $1$ .

$u_{upper} = 0$

Step 1.6

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

$u_{lower} = 2$

$u_{upper} = 0$

Step 1.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\int_{2}^{0} {u_{2}}^{2} \frac{1}{- 7} d u_{2}$

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

$\int_{2}^{0} {u_{2}}^{2} (- \frac{1}{7}) d u_{2}$

Step 2.2

Combine ${u_{2}}^{2}$ and $\frac{1}{7}$ .

$\int_{2}^{0} - \frac{{u_{2}}^{2}}{7} d u_{2}$

Step 3

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$- \int_{2}^{0} \frac{{u_{2}}^{2}}{7} d u_{2}$

Step 4

Since $\frac{1}{7}$ is constant with respect to $u_{2}$ , move $\frac{1}{7}$ out of the integral.

$- (\frac{1}{7} \int_{2}^{0} {u_{2}}^{2} d u_{2})$

Step 5

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

$- \frac{1}{7} \frac{1}{3} {u_{2}}^{3}]_{2}^{0}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{1}{3} {u_{2}}^{3}$ at $0$ and at $2$ .

$- \frac{1}{7} ((\frac{1}{3} \cdot 0^{3}) - \frac{1}{3} \cdot 2^{3})$

Step 6.2

Simplify.

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Step 6.2.1

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

$- \frac{1}{7} (\frac{1}{3} \cdot 0 - \frac{1}{3} \cdot 2^{3})$

Step 6.2.2

Multiply $\frac{1}{3}$ by $0$ .

$- \frac{1}{7} (0 - \frac{1}{3} \cdot 2^{3})$

Step 6.2.3

Raise $2$ to the power of $3$ .

$- \frac{1}{7} (0 - \frac{1}{3} \cdot 8)$

Step 6.2.4

Multiply $8$ by $- 1$ .

$- \frac{1}{7} (0 - 8 (\frac{1}{3}))$

Step 6.2.5

Combine $- 8$ and $\frac{1}{3}$ .

$- \frac{1}{7} (0 + \frac{- 8}{3})$

Step 6.2.6

Move the negative in front of the fraction.

$- \frac{1}{7} (0 - \frac{8}{3})$

Step 6.2.7

Subtract $\frac{8}{3}$ from $0$ .

$- \frac{1}{7} (- \frac{8}{3})$

Step 6.2.8

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{7}) \frac{8}{3}$

Step 6.2.9

Multiply $\frac{1}{7}$ by $1$ .

$\frac{1}{7} \cdot \frac{8}{3}$

Step 6.2.10

Multiply $\frac{1}{7}$ by $\frac{8}{3}$ .

$\frac{8}{7 \cdot 3}$

Step 6.2.11

Multiply $7$ by $3$ .

$\frac{8}{21}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{8}{21}$

Decimal Form:

$0.\overline{380952}$