Calculus Examples

$e^{x} (2 + \frac{e^{- x}}{\cos^{2} (x)})$

Step 1

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

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

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.

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply by $1$ .

$\int e^{x} (2 + \frac{e^{- x}}{\cos^{2} (x) \cdot 1}) d x$

Step 4.1.2

Separate fractions.

$\int e^{x} (2 + \frac{e^{- x}}{1} \cdot \frac{1}{\cos^{2} (x)}) d x$

Step 4.1.3

Convert from $\frac{1}{\cos^{2} (x)}$ to $\sec^{2} (x)$ .

$\int e^{x} (2 + \frac{e^{- x}}{1} \sec^{2} (x)) d x$

Step 4.1.4

Divide $e^{- x}$ by $1$ .

$\int e^{x} (2 + e^{- x} \sec^{2} (x)) d x$

Step 4.2

Apply the distributive property.

$\int e^{x} \cdot 2 + e^{x} (e^{- x} \sec^{2} (x)) d x$

Step 4.3

Move $2$ to the left of $e^{x}$ .

$\int 2 \cdot e^{x} + e^{x} (e^{- x} \sec^{2} (x)) d x$

Step 4.4

Multiply $e^{x}$ by $e^{- x}$ by adding the exponents.

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Step 4.4.1

Move $e^{- x}$ .

$\int 2 \cdot e^{x} + e^{- x} e^{x} \sec^{2} (x) d x$

Step 4.4.2

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

$\int 2 \cdot e^{x} + e^{- x + x} \sec^{2} (x) d x$

Step 4.4.3

Add $- x$ and $x$ .

$\int 2 \cdot e^{x} + e^{0} \sec^{2} (x) d x$

Step 4.5

Simplify $e^{0} \sec^{2} (x)$ .

$\int 2 e^{x} + \sec^{2} (x) d x$

Step 5

Split the single integral into multiple integrals.

$\int 2 e^{x} d x + \int \sec^{2} (x) d x$

Step 6

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int e^{x} d x + \int \sec^{2} (x) d x$

Step 7

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

$2 (e^{x} + C) + \int \sec^{2} (x) d x$

Step 8

Since the derivative of $\tan (x)$ is $\sec^{2} (x)$ , the integral of $\sec^{2} (x)$ is $\tan (x)$ .

$2 (e^{x} + C) + \tan (x) + C$

Step 9

Simplify.

$2 e^{x} + \tan (x) + C$

Step 10

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

$F (x) =$ $2 e^{x} + \tan (x) + C$