Calculus Examples

\int_{0}^{\frac{π}{2}} e^{\sin (7 π x)} \cos (7 π x) d x

$\int_{0}^{\frac{π}{2}} e^{\sin (7 π x)} \cos (7 π x) d x$

Step 1

Let

u_{2} = \sin (7 π x)

$u_{2} = \sin (7 π x)$ . Then

d u_{2} = 7 π \cos (7 π x) d x

$d u_{2} = 7 π \cos (7 π x) d x$ , so

\frac{1}{7 π} d u_{2} = \cos (7 π x) d x

$\frac{1}{7 π} d u_{2} = \cos (7 π x) d x$ . Rewrite using

u_{2}

$u_{2}$ and

d

$d$

u_{2}

$u_{2}$ .

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Step 1.1

Let

u_{2} = \sin (7 π x)

$u_{2} = \sin (7 π x)$ . Find

\frac{d u_{2}}{d x}

$\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate

\sin (7 π x)

$\sin (7 π x)$ .

\frac{d}{d x} [\sin (7 π x)]

$\frac{d}{d x} [\sin (7 π x)]$

Step 1.1.2

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ where

f (x) = \sin (x)

$f (x) = \sin (x)$ and

g (x) = 7 π x

$g (x) = 7 π x$ .

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Step 1.1.2.1

To apply the Chain Rule, set

u_{1}

$u_{1}$ as

7 π x

$7 π x$ .

\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [7 π x]

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [7 π x]$

Step 1.1.2.2

The derivative of

\sin (u_{1})

$\sin (u_{1})$ with respect to

u_{1}

$u_{1}$ is

\cos (u_{1})

$\cos (u_{1})$ .

\cos (u_{1}) \frac{d}{d x} [7 π x]

$\cos (u_{1}) \frac{d}{d x} [7 π x]$

Step 1.1.2.3

Replace all occurrences of

u_{1}

$u_{1}$ with

7 π x

$7 π x$ .

\cos (7 π x) \frac{d}{d x} [7 π x]

$\cos (7 π x) \frac{d}{d x} [7 π x]$

\cos (7 π x) \frac{d}{d x} [7 π x]

$\cos (7 π x) \frac{d}{d x} [7 π x]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Since

7 π

$7 π$ is constant with respect to

x

$x$ , the derivative of

7 π x

$7 π x$ with respect to

x

$x$ is

7 π \frac{d}{d x} [x]

$7 π \frac{d}{d x} [x]$ .

\cos (7 π x) (7 π \frac{d}{d x} [x])

$\cos (7 π x) (7 π \frac{d}{d x} [x])$

Step 1.1.3.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

\cos (7 π x) (7 π \cdot 1)

$\cos (7 π x) (7 π \cdot 1)$

Step 1.1.3.3

Simplify the expression.

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Step 1.1.3.3.1

Multiply

7

$7$ by

1

$1$ .

\cos (7 π x) (7 π)

$\cos (7 π x) (7 π)$

Step 1.1.3.3.2

Move

7

$7$ to the left of

\cos (7 π x)

$\cos (7 π x)$ .

7 \cdot \cos (7 π x) π

$7 \cdot \cos (7 π x) π$

Step 1.1.3.3.3

Reorder the factors of

7 \cos (7 π x) π

$7 \cos (7 π x) π$ .

7 π \cos (7 π x)

$7 π \cos (7 π x)$

7 π \cos (7 π x)

$7 π \cos (7 π x)$

7 π \cos (7 π x)

$7 π \cos (7 π x)$

7 π \cos (7 π x)

$7 π \cos (7 π x)$

Step 1.2

Substitute the lower limit in for

x

$x$ in

u_{2} = \sin (7 π x)

$u_{2} = \sin (7 π x)$ .

u_{lower} = \sin (7 π \cdot 0)

$u_{lower} = \sin (7 π \cdot 0)$

Step 1.3

Simplify.

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Step 1.3.1

Multiply

7 π \cdot 0

$7 π \cdot 0$ .

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Step 1.3.1.1

Multiply

0

$0$ by

7

$7$ .

u_{lower} = \sin (0 π)

$u_{lower} = \sin (0 π)$

Step 1.3.1.2

Multiply

0

$0$ by

π

$π$ .

u_{lower} = \sin (0)

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

u_{lower} = \sin (0)

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

Step 1.3.2

The exact value of

\sin (0)

$\sin (0)$ is

0

$0$ .

u_{lower} = 0

$u_{lower} = 0$

u_{lower} = 0

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for

x

$x$ in

u_{2} = \sin (7 π x)

$u_{2} = \sin (7 π x)$ .

u_{upper} = \sin (7 π \frac{π}{2})

$u_{upper} = \sin (7 π \frac{π}{2})$

Step 1.5

Simplify.

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Step 1.5.1

Multiply

7 π \frac{π}{2}

$7 π \frac{π}{2}$ .

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Step 1.5.1.1

Combine

\frac{π}{2}

$\frac{π}{2}$ and

7

$7$ .

u_{upper} = \sin (\frac{π \cdot 7}{2} π)

$u_{upper} = \sin (\frac{π \cdot 7}{2} π)$

Step 1.5.1.2

Combine

\frac{π \cdot 7}{2}

$\frac{π \cdot 7}{2}$ and

π

$π$ .

u_{upper} = \sin (\frac{π \cdot 7 π}{2})

$u_{upper} = \sin (\frac{π \cdot 7 π}{2})$

Step 1.5.1.3

Raise

π

$π$ to the power of

1

$1$ .

u_{upper} = \sin (\frac{7 (π^{1} π)}{2})

$u_{upper} = \sin (\frac{7 (π^{1} π)}{2})$

Step 1.5.1.4

Raise

π

$π$ to the power of

1

$1$ .

u_{upper} = \sin (\frac{7 (π^{1} π^{1})}{2})

$u_{upper} = \sin (\frac{7 (π^{1} π^{1})}{2})$

Step 1.5.1.5

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

u_{upper} = \sin (\frac{7 π^{1 + 1}}{2})

$u_{upper} = \sin (\frac{7 π^{1 + 1}}{2})$

Step 1.5.1.6

Add

1

$1$ and

1

$1$ .

u_{upper} = \sin (\frac{7 π^{2}}{2})

$u_{upper} = \sin (\frac{7 π^{2}}{2})$

u_{upper} = \sin (\frac{7 π^{2}}{2})

$u_{upper} = \sin (\frac{7 π^{2}}{2})$

Step 1.5.2

Evaluate

\sin (\frac{7 π^{2}}{2})

$\sin (\frac{7 π^{2}}{2})$ .

u_{upper} = 0.01390333

$u_{upper} = 0.01390333$

u_{upper} = 0.01390333

$u_{upper} = 0.01390333$

Step 1.6

The values found for

u_{lower}

$u_{lower}$ and

u_{upper}

$u_{upper}$ will be used to evaluate the definite integral.

u_{lower} = 0

$u_{lower} = 0$

u_{upper} = 0.01390333

$u_{upper} = 0.01390333$

Step 1.7

Rewrite the problem using

u_{2}

$u_{2}$ ,

d u_{2}

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

\int_{0}^{0.01390333} e^{u_{2}} \frac{1}{7 π} d u_{2}

$\int_{0}^{0.01390333} e^{u_{2}} \frac{1}{7 π} d u_{2}$

\int_{0}^{0.01390333} e^{u_{2}} \frac{1}{7 π} d u_{2}

$\int_{0}^{0.01390333} e^{u_{2}} \frac{1}{7 π} d u_{2}$

Step 2

Combine

e^{u_{2}}

$e^{u_{2}}$ and

\frac{1}{7 π}

$\frac{1}{7 π}$ .

\int_{0}^{0.01390333} \frac{e^{u_{2}}}{7 π} d u_{2}

$\int_{0}^{0.01390333} \frac{e^{u_{2}}}{7 π} d u_{2}$

Step 3

Since

\frac{1}{7 π}

$\frac{1}{7 π}$ is constant with respect to

u_{2}

$u_{2}$ , move

\frac{1}{7 π}

$\frac{1}{7 π}$ out of the integral.

\frac{1}{7 π} \int_{0}^{0.01390333} e^{u_{2}} d u_{2}

$\frac{1}{7 π} \int_{0}^{0.01390333} e^{u_{2}} d u_{2}$

Step 4

The integral of

e^{u_{2}}

$e^{u_{2}}$ with respect to

u_{2}

$u_{2}$ is

e^{u_{2}}

$e^{u_{2}}$ .

\frac{1}{7 π} e^{u_{2}}]_{0}^{0.01390333}

$\frac{1}{7 π} e^{u_{2}}]_{0}^{0.01390333}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate

e^{u_{2}}

$e^{u_{2}}$ at

0.01390333

$0.01390333$ and at

0

$0$ .

\frac{1}{7 π} ((e^{0.01390333}) - e^{0})

$\frac{1}{7 π} ((e^{0.01390333}) - e^{0})$

Step 5.2

Simplify.

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Step 5.2.1

Anything raised to

0

$0$ is

1

$1$ .

\frac{1}{7 π} (e^{0.01390333} - 1 \cdot 1)

$\frac{1}{7 π} (e^{0.01390333} - 1 \cdot 1)$

Step 5.2.2

Multiply

- 1

$- 1$ by

1

$1$ .

\frac{1}{7 π} (e^{0.01390333} - 1)

$\frac{1}{7 π} (e^{0.01390333} - 1)$

\frac{1}{7 π} (e^{0.01390333} - 1)

$\frac{1}{7 π} (e^{0.01390333} - 1)$

\frac{1}{7 π} (e^{0.01390333} - 1)

$\frac{1}{7 π} (e^{0.01390333} - 1)$

Step 6

The result can be shown in multiple forms.

Exact Form:

\frac{1}{7 π} \cdot (e^{0.01390333} - 1)

$\frac{1}{7 π} \cdot (e^{0.01390333} - 1)$

Decimal Form:

0.00063663 \dots

$0.00063663 \dots$