Calculus Examples

$c' (x) = x^{\frac{1}{2} + 3}$

Step 1

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

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

Step 2

Simplify.

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Step 2.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.2

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

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Simplify the numerator.

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Step 2.4.1

Multiply $3$ by $2$ .

$\int x^{\frac{1 + 6}{2}} d x$

Step 2.4.2

Add $1$ and $6$ .

$\int x^{\frac{7}{2}} d x$

Step 3

By the Power Rule, the integral of $x^{\frac{7}{2}}$ with respect to $x$ is $\frac{2}{9} x^{\frac{9}{2}}$ .

$\frac{2}{9} x^{\frac{9}{2}} + C$

Step 4

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

$c (x) = \frac{2}{9} x^{\frac{9}{2}} + C$