Calculus Examples

$\int e^{\cos (π x)} \sin (π x) d x$

Step 1

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

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Step 1.1

Let $u_{2} = \cos (π x)$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $\cos (π x)$ .

$\frac{d}{d x} [\cos (π x)]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = π x$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $π x$ .

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

Step 1.1.2.2

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

$- \sin (u_{1}) \frac{d}{d x} [π x]$

Step 1.1.2.3

Replace all occurrences of $u_{1}$ with $π x$ .

$- \sin (π x) \frac{d}{d x} [π x]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

$- \sin (π x) (π \frac{d}{d x} [x])$

Step 1.1.3.2

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

$- \sin (π x) (π \cdot 1)$

Step 1.1.3.3

Simplify the expression.

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Step 1.1.3.3.1

Multiply $π$ by $1$ .

$- \sin (π x) π$

Step 1.1.3.3.2

Reorder the factors of $- \sin (π x) π$ .

$- π \sin (π x)$

Step 1.2

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

$\int - e^{u_{2}} \frac{- 1}{- π} d u_{2}$

Step 2

Simplify.

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Step 2.1

Dividing two negative values results in a positive value.

$\int - e^{u_{2}} \frac{1}{π} d u_{2}$

Step 2.2

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

$\int - \frac{e^{u_{2}}}{π} d u_{2}$

Step 3

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

$- \int \frac{e^{u_{2}}}{π} d u_{2}$

Step 4

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

$- (\frac{1}{π} \int e^{u_{2}} d u_{2})$

Step 5

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$- \frac{1}{π} (e^{u_{2}} + C)$

Step 6

Simplify.

$- \frac{1}{π} e^{u_{2}} + C$

Step 7

Replace all occurrences of $u_{2}$ with $\cos (π x)$ .

$- \frac{1}{π} e^{\cos (π x)} + C$