Calculus Examples

$\int \frac{5 y - 10}{y^{2} - 4 y + 11} d y$

Step 1

Factor $5$ out of $5 y - 10$ .

Tap for more steps...

Step 1.1

Factor $5$ out of $5 y$ .

$\int \frac{5 (y) - 10}{y^{2} - 4 y + 11} d y$

Step 1.2

Factor $5$ out of $- 10$ .

$\int \frac{5 y + 5 \cdot - 2}{y^{2} - 4 y + 11} d y$

Step 1.3

Factor $5$ out of $5 y + 5 \cdot - 2$ .

$\int \frac{5 (y - 2)}{y^{2} - 4 y + 11} d y$

Step 2

Let $u = y^{2} - 4 y + 11$ . Then $d u = (2 y - 4) d y$ , so $\frac{1}{2} d u = (y - 2) d y$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.1

Let $u = y^{2} - 4 y + 11$ . Find $\frac{d u}{d y}$ .

Tap for more steps...

Step 2.1.1

Differentiate $y^{2} - 4 y + 11$ .

$\frac{d}{d y} [y^{2} - 4 y + 11]$

Step 2.1.2

Differentiate.

Tap for more steps...

Step 2.1.2.1

By the Sum Rule, the derivative of $y^{2} - 4 y + 11$ with respect to $y$ is $\frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 4 y] + \frac{d}{d y} [11]$ .

$\frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 4 y] + \frac{d}{d y} [11]$

Step 2.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$2 y + \frac{d}{d y} [- 4 y] + \frac{d}{d y} [11]$

Step 2.1.3

Evaluate $\frac{d}{d y} [- 4 y]$ .

Tap for more steps...

Step 2.1.3.1

Since $- 4$ is constant with respect to $y$ , the derivative of $- 4 y$ with respect to $y$ is $- 4 \frac{d}{d y} [y]$ .

$2 y - 4 \frac{d}{d y} [y] + \frac{d}{d y} [11]$

Step 2.1.3.2

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

$2 y - 4 \cdot 1 + \frac{d}{d y} [11]$

Step 2.1.3.3

Multiply $- 4$ by $1$ .

$2 y - 4 + \frac{d}{d y} [11]$

Step 2.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.1.4.1

Since $11$ is constant with respect to $y$ , the derivative of $11$ with respect to $y$ is $0$ .

$2 y - 4 + 0$

Step 2.1.4.2

Add $2 y - 4$ and $0$ .

$2 y - 4$

Step 2.2

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

$\int \frac{5}{u} \cdot \frac{1}{2} d u$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Multiply $\frac{5}{u}$ by $\frac{1}{2}$ .

$\int \frac{5}{u \cdot 2} d u$

Step 3.2

Move $2$ to the left of $u$ .

$\int \frac{5}{2 u} d u$

Step 4

Since $\frac{5}{2}$ is constant with respect to $u$ , move $\frac{5}{2}$ out of the integral.

$\frac{5}{2} \int \frac{1}{u} d u$

Step 5

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$\frac{5}{2} (\ln (| u |) + C)$

Step 6

Simplify.

$\frac{5}{2} \ln (| u |) + C$

Step 7

Replace all occurrences of $u$ with $y^{2} - 4 y + 11$ .

$\frac{5}{2} \ln (| y^{2} - 4 y + 11 |) + C$