Calculus Examples

$\int - \frac{2200 x}{\sqrt{25 - x^{2}}} d x$

Step 1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \int \frac{2200 x}{\sqrt{25 - x^{2}}} d x$

Step 2

Since $2200$ is constant with respect to $x$ , move $2200$ out of the integral.

$- (2200 \int \frac{x}{\sqrt{25 - x^{2}}} d x)$

Step 3

Multiply $2200$ by $- 1$ .

$- 2200 \int \frac{x}{\sqrt{25 - x^{2}}} d x$

Step 4

Let $u = 25 - x^{2}$ . Then $d u = - 2 x d x$ , so $- \frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 4.1

Let $u = 25 - x^{2}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 4.1.1

Differentiate $25 - x^{2}$ .

$\frac{d}{d x} [25 - x^{2}]$

Step 4.1.2

Differentiate.

Tap for more steps...

Step 4.1.2.1

By the Sum Rule, the derivative of $25 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [25] + \frac{d}{d x} [- x^{2}]$ .

$\frac{d}{d x} [25] + \frac{d}{d x} [- x^{2}]$

Step 4.1.2.2

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x^{2}]$

Step 4.1.3

Evaluate $\frac{d}{d x} [- x^{2}]$ .

Tap for more steps...

Step 4.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$0 - \frac{d}{d x} [x^{2}]$

Step 4.1.3.2

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

$0 - (2 x)$

Step 4.1.3.3

Multiply $2$ by $- 1$ .

$0 - 2 x$

Step 4.1.4

Subtract $2 x$ from $0$ .

$- 2 x$

Step 4.2

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

$- 2200 \int \frac{1}{\sqrt{u}} \cdot \frac{1}{- 2} d u$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Move the negative in front of the fraction.

$- 2200 \int \frac{1}{\sqrt{u}} (- \frac{1}{2}) d u$

Step 5.2

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

$- 2200 \int - \frac{1}{\sqrt{u} \cdot 2} d u$

Step 5.3

Move $2$ to the left of $\sqrt{u}$ .

$- 2200 \int - \frac{1}{2 \sqrt{u}} d u$

Step 6

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- 2200 (- \int \frac{1}{2 \sqrt{u}} d u)$

Step 7

Multiply $- 1$ by $- 2200$ .

$2200 \int \frac{1}{2 \sqrt{u}} d u$

Step 8

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$2200 (\frac{1}{2} \int \frac{1}{\sqrt{u}} d u)$

Step 9

Simplify the expression.

Tap for more steps...

Step 9.1

Simplify.

Tap for more steps...

Step 9.1.1

Combine $\frac{1}{2}$ and $2200$ .

$\frac{2200}{2} \int \frac{1}{\sqrt{u}} d u$

Step 9.1.2

Cancel the common factor of $2200$ and $2$ .

Tap for more steps...

Step 9.1.2.1

Factor $2$ out of $2200$ .

$\frac{2 \cdot 1100}{2} \int \frac{1}{\sqrt{u}} d u$

Step 9.1.2.2

Cancel the common factors.

Tap for more steps...

Step 9.1.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 1100}{2 (1)} \int \frac{1}{\sqrt{u}} d u$

Step 9.1.2.2.2

Cancel the common factor.

$\frac{2 \cdot 1100}{2 \cdot 1} \int \frac{1}{\sqrt{u}} d u$

Step 9.1.2.2.3

Rewrite the expression.

$\frac{1100}{1} \int \frac{1}{\sqrt{u}} d u$

Step 9.1.2.2.4

Divide $1100$ by $1$ .

$1100 \int \frac{1}{\sqrt{u}} d u$

Step 9.2

Apply basic rules of exponents.

Tap for more steps...

Step 9.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$1100 \int \frac{1}{u^{\frac{1}{2}}} d u$

Step 9.2.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$1100 \int {(u^{\frac{1}{2}})}^{- 1} d u$

Step 9.2.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

Tap for more steps...

Step 9.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$1100 \int u^{\frac{1}{2} \cdot - 1} d u$

Step 9.2.3.2

Combine $\frac{1}{2}$ and $- 1$ .

$1100 \int u^{\frac{- 1}{2}} d u$

Step 9.2.3.3

Move the negative in front of the fraction.

$1100 \int u^{- \frac{1}{2}} d u$

Step 10

By the Power Rule, the integral of $u^{- \frac{1}{2}}$ with respect to $u$ is $2 u^{\frac{1}{2}}$ .

$1100 (2 u^{\frac{1}{2}} + C)$

Step 11

Simplify.

Tap for more steps...

Step 11.1

Rewrite $1100 (2 u^{\frac{1}{2}} + C)$ as $1100 \cdot 2 u^{\frac{1}{2}} + C$ .

$1100 \cdot 2 u^{\frac{1}{2}} + C$

Step 11.2

Multiply $1100$ by $2$ .

$2200 u^{\frac{1}{2}} + C$

Step 12

Replace all occurrences of $u$ with $25 - x^{2}$ .

$2200 {(25 - x^{2})}^{\frac{1}{2}} + C$