Algebra Examples

$f (x) = x \sqrt{x^{2}} + 1$

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

Simplify.

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Step 2.1

Pull terms out from under the radical, assuming positive real numbers.

$\int x \cdot x + 1 d x$

Step 2.2

Raise $x$ to the power of $1$ .

$\int x^{1} x + 1 d x$

Step 2.3

Raise $x$ to the power of $1$ .

$\int x^{1} x^{1} + 1 d x$

Step 2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{1 + 1} + 1 d x$

Step 2.5

Add $1$ and $1$ .

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

Step 3

Split the single integral into multiple integrals.

$\int x^{2} d x + \int d x$

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

Simplify.

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

Step 7

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

$F (x) = \frac{1}{3} x^{3} + x + C$