Calculus Examples

\int \cos (x y) d x

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

Step 1

Let

u = x y

$u = x y$ . Then

d u = y d x

$d u = y d x$ , so

\frac{1}{y} d u = d x

$\frac{1}{y} d u = d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 1.1

Let

u = x y

$u = x y$ . Find

\frac{d u}{d x}

$\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate

x y

$x y$ .

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

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

Step 1.1.2

Since

y

$y$ is constant with respect to

x

$x$ , the derivative of

x y

$x y$ with respect to

x

$x$ is

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

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

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}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

y \cdot 1

$y \cdot 1$

Step 1.1.4

Multiply

y

$y$ by

1

$1$ .

y

$y$

y

$y$

Step 1.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

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

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

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

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

Step 2

Combine

\cos (u)

$\cos (u)$ and

\frac{1}{y}

$\frac{1}{y}$ .

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

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

Step 3

Since

\frac{1}{y}

$\frac{1}{y}$ is constant with respect to

u

$u$ , move

\frac{1}{y}

$\frac{1}{y}$ out of the integral.

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

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

Step 4

The integral of

\cos (u)

$\cos (u)$ with respect to

u

$u$ is

\sin (u)

$\sin (u)$ .

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

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

Step 5

Simplify.

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Step 5.1

Simplify.

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

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

Step 5.2

Combine

\frac{1}{y}

$\frac{1}{y}$ and

\sin (u)

$\sin (u)$ .

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

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

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

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

Step 6

Replace all occurrences of

u

$u$ with

x y

$x y$ .

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

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

\int_{}^{} \cos (x y) d x

$\int_{}^{} \cos (x y) d x$

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[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$