Calculus Examples

$f' (x) = 1 + 3 \sqrt{x}$

Step 1

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

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

Step 2

Split the single integral into multiple integrals.

$\int d x + \int 3 \sqrt{x} d x$

Step 3

Apply the constant rule.

$x + C + \int 3 \sqrt{x} d x$

Step 4

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$x + C + 3 \int \sqrt{x} d x$

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Simplify.

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

Step 7.2

Simplify.

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Step 7.2.1

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

$x + \frac{3 \cdot 2}{3} x^{\frac{3}{2}} + C$

Step 7.2.2

Multiply $3$ by $2$ .

$x + \frac{6}{3} x^{\frac{3}{2}} + C$

Step 7.2.3

Cancel the common factor of $6$ and $3$ .

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Step 7.2.3.1

Factor $3$ out of $6$ .

$x + \frac{3 \cdot 2}{3} x^{\frac{3}{2}} + C$

Step 7.2.3.2

Cancel the common factors.

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Step 7.2.3.2.1

Factor $3$ out of $3$ .

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

Step 7.2.3.2.2

Cancel the common factor.

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

Step 7.2.3.2.3

Rewrite the expression.

$x + \frac{2}{1} x^{\frac{3}{2}} + C$

Step 7.2.3.2.4

Divide $2$ by $1$ .

$x + 2 x^{\frac{3}{2}} + C$

Step 8

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

$f (x) = x + 2 x^{\frac{3}{2}} + C$