Calculus Examples

$\int \cos (x y) d x$

Step 1

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

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Differentiate $x y$ .

$\frac{d}{d x} [x y]$

Step 1.1.2

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

$y \frac{d}{d x} [x]$

Step 1.1.3

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

$y \cdot 1$

Step 1.1.4

Multiply $y$ by $1$ .

$y$

Step 1.2

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

$\int \cos (u) \frac{1}{y} d u$

Step 2

Combine $\cos (u)$ and $\frac{1}{y}$ .

$\int \frac{\cos (u)}{y} d u$

Step 3

Since $\frac{1}{y}$ is constant with respect to $u$ , move $\frac{1}{y}$ out of the integral.

$\frac{1}{y} \int \cos (u) d u$

Step 4

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

$\frac{1}{y} (\sin (u) + C)$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Simplify.

$\frac{1}{y} \sin (u) + C$

Step 5.2

Combine $\frac{1}{y}$ and $\sin (u)$ .

$\frac{\sin (u)}{y} + C$

Step 6

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

$\frac{\sin (x y)}{y} + C$