Calculus Examples

$\int_{1}^{2} y \sin (x y) d x$

Step 1

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

$y \int_{1}^{2} \sin (x y) d x$

Step 2

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

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Step 2.1

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

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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

Substitute the lower limit in for $x$ in $u = x y$ .

$u_{lower} = 1 y$

Step 2.3

Multiply $y$ by $1$ .

$u_{lower} = y$

Step 2.4

Substitute the upper limit in for $x$ in $u = x y$ .

$u_{upper} = 2 y$

Step 2.5

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = y$

$u_{upper} = 2 y$

Step 2.6

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$y \int_{y}^{2 y} \sin (u) \frac{1}{y} d u$

Step 3

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

$y \int_{y}^{2 y} \frac{\sin (u)}{y} d u$

Step 4

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

$y (\frac{1}{y} \int_{y}^{2 y} \sin (u) d u)$

Step 5

Simplify.

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Step 5.1

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

$\frac{y}{y} \int_{y}^{2 y} \sin (u) d u$

Step 5.2

Cancel the common factor of $y$ .

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Step 5.2.1

Cancel the common factor.

$\frac{y}{y} \int_{y}^{2 y} \sin (u) d u$

Step 5.2.2

Rewrite the expression.

$1 \int_{y}^{2 y} \sin (u) d u$

Step 5.3

Multiply $\int_{y}^{2 y} \sin (u) d u$ by $1$ .

$\int_{y}^{2 y} \sin (u) d u$

Step 6

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

$- \cos (u)]_{y}^{2 y}$

Step 7

Evaluate $- \cos (u)$ at $2 y$ and at $y$ .

$- \cos (2 y) + \cos (y)$