Calculus Examples

$\int y e^{x y} d x$

Step 1

Since $y$ is constant with respect to $x$ , move $y$ out of the integral.

$y \int e^{x y} d x$

Step 2

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 2.1

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

Tap for more steps...

Step 2.1.1

Differentiate $x y$ .

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

Step 2.1.2

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 2.1.3

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 2.1.4

Multiply $y$ by $1$ .

$y$

Step 2.2

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

$y \int e^{u} \frac{1}{y} d u$

Step 3

Combine $e^{u}$ and $\frac{1}{y}$ .

$y \int \frac{e^{u}}{y} d u$

Step 4

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

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Combine $\frac{1}{y}$ and $y$ .

$\frac{y}{y} \int e^{u} d u$

Step 5.2

Cancel the common factor of $y$ .

Tap for more steps...

Step 5.2.1

Cancel the common factor.

$\frac{y}{y} \int e^{u} d u$

Step 5.2.2

Rewrite the expression.

$1 \int e^{u} d u$

Step 5.3

Multiply $\int e^{u} d u$ by $1$ .

$\int e^{u} d u$

Step 6

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

$e^{u} + C$

Step 7

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

$e^{x y} + C$