Calculus Examples

$R' (x) = 4 x {(x^{2} + 27000)}^{- \frac{2}{3}}$

Step 1

The function $R (x)$ can be found by evaluating the indefinite integral of the derivative $R' (x)$ .

$R (x) = \int R' (x) d x$

Step 2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int x {(x^{2} + 27000)}^{- \frac{2}{3}} d x$

Step 3

Let $u = x^{2} + 27000$ . 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 3.1

Let $u = x^{2} + 27000$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 3.1.1

Differentiate $x^{2} + 27000$ .

$\frac{d}{d x} [x^{2} + 27000]$

Step 3.1.2

By the Sum Rule, the derivative of $x^{2} + 27000$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [27000]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [27000]$

Step 3.1.3

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

$2 x + \frac{d}{d x} [27000]$

Step 3.1.4

Since $27000$ is constant with respect to $x$ , the derivative of $27000$ with respect to $x$ is $0$ .

$2 x + 0$

Step 3.1.5

Add $2 x$ and $0$ .

$2 x$

Step 3.2

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

$4 \int u^{- \frac{2}{3}} \frac{1}{2} d u$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Combine $u^{- \frac{2}{3}}$ and $\frac{1}{2}$ .

$4 \int \frac{u^{- \frac{2}{3}}}{2} d u$

Step 4.2

Move $u^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 \int \frac{1}{2 u^{\frac{2}{3}}} d u$

Step 5

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

$4 (\frac{1}{2} \int \frac{1}{u^{\frac{2}{3}}} d u)$

Step 6

Simplify the expression.

Tap for more steps...

Step 6.1

Simplify.

Tap for more steps...

Step 6.1.1

Combine $\frac{1}{2}$ and $4$ .

$\frac{4}{2} \int \frac{1}{u^{\frac{2}{3}}} d u$

Step 6.1.2

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 6.1.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2} \int \frac{1}{u^{\frac{2}{3}}} d u$

Step 6.1.2.2

Cancel the common factors.

Tap for more steps...

Step 6.1.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 2}{2 (1)} \int \frac{1}{u^{\frac{2}{3}}} d u$

Step 6.1.2.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 1} \int \frac{1}{u^{\frac{2}{3}}} d u$

Step 6.1.2.2.3

Rewrite the expression.

$\frac{2}{1} \int \frac{1}{u^{\frac{2}{3}}} d u$

Step 6.1.2.2.4

Divide $2$ by $1$ .

$2 \int \frac{1}{u^{\frac{2}{3}}} d u$

Step 6.2

Apply basic rules of exponents.

Tap for more steps...

Step 6.2.1

Move $u^{\frac{2}{3}}$ out of the denominator by raising it to the $- 1$ power.

$2 \int {(u^{\frac{2}{3}})}^{- 1} d u$

Step 6.2.2

Multiply the exponents in ${(u^{\frac{2}{3}})}^{- 1}$ .

Tap for more steps...

Step 6.2.2.1

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

$2 \int u^{\frac{2}{3} \cdot - 1} d u$

Step 6.2.2.2

Multiply $\frac{2}{3} \cdot - 1$ .

Tap for more steps...

Step 6.2.2.2.1

Combine $\frac{2}{3}$ and $- 1$ .

$2 \int u^{\frac{2 \cdot - 1}{3}} d u$

Step 6.2.2.2.2

Multiply $2$ by $- 1$ .

$2 \int u^{\frac{- 2}{3}} d u$

Step 6.2.2.3

Move the negative in front of the fraction.

$2 \int u^{- \frac{2}{3}} d u$

Step 7

By the Power Rule, the integral of $u^{- \frac{2}{3}}$ with respect to $u$ is $3 u^{\frac{1}{3}}$ .

$2 (3 u^{\frac{1}{3}} + C)$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Rewrite $2 (3 u^{\frac{1}{3}} + C)$ as $2 \cdot 3 u^{\frac{1}{3}} + C$ .

$2 \cdot 3 u^{\frac{1}{3}} + C$

Step 8.2

Multiply $2$ by $3$ .

$6 u^{\frac{1}{3}} + C$

Step 9

Replace all occurrences of $u$ with $x^{2} + 27000$ .

$6 {(x^{2} + 27000)}^{\frac{1}{3}} + C$

Step 10

The function $R$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$R (x) = 6 {(x^{2} + 27000)}^{\frac{1}{3}} + C$