Calculus Examples

${(e^{\frac{x}{2}})}^{2}$

Step 1

Write ${(e^{\frac{x}{2}})}^{2}$ as a function.

$f (x) = {(e^{\frac{x}{2}})}^{2}$

Step 2

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

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

Step 3

Set up the integral to solve.

$F (x) = \int {(e^{\frac{x}{2}})}^{2} d x$

Step 4

Multiply the exponents in ${(e^{\frac{x}{2}})}^{2}$ .

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Step 4.1

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

$\int e^{\frac{x}{2} \cdot 2} d x$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Cancel the common factor.

$\int e^{\frac{x}{2} \cdot 2} d x$

Step 4.2.2

Rewrite the expression.

$\int e^{x} d x$

Step 5

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$e^{x} + C$

Step 6

The answer is the antiderivative of the function $f (x) = {(e^{\frac{x}{2}})}^{2}$ .

$F (x) =$ $e^{x} + C$