Calculus Examples

$\int π \sin (π x) d x$

Step 1

Let $u = π x$ . Then $d u = π d x$ , so $\frac{1}{π} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = π x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $π x$ .

$\frac{d}{d x} [π x]$

Step 1.1.2

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

$π \frac{d}{d x} [x]$

Step 1.1.3

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

$π \cdot 1$

Step 1.1.4

Multiply $π$ by $1$ .

$π$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int \sin (u) d u$

Step 2

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

$- \cos (u) + C$

Step 3

Replace all occurrences of $u$ with $π x$ .

$- \cos (π x) + C$