Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

Apply the constant rule.

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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Step 4.1.1

Differentiate $x y$ .

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

Step 4.1.2

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 4.1.3

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 4.1.4

Multiply $y$ by $1$ .

$y$

Step 4.2

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Cancel the common factor of $y$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

$y x + C + 1 \int \cos (u) d u$

Step 7.3

Multiply $\int \cos (u) d u$ by $1$ .

$y x + C + \int \cos (u) d u$

Step 8

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

$y x + C + \sin (u) + C$

Step 9

Simplify.

$y x + \sin (u) + C$

Step 10

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

$y x + \sin (x y) + C$