Calculus Examples

$e^{x} (1 - 3 e^{- 2 x})$

Step 1

Write $e^{x} (1 - 3 e^{- 2 x})$ as a function.

$f (x) = e^{x} (1 - 3 e^{- 2 x})$

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.

$F (x) = \int e^{x} (1 - 3 e^{- 2 x}) d x$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\int e^{x} \cdot 1 + e^{x} (- 3 e^{- 2 x}) d x$

Step 4.2

Multiply $e^{x}$ by $1$ .

$\int e^{x} + e^{x} (- 3 e^{- 2 x}) d x$

Step 4.3

Rewrite using the commutative property of multiplication.

$\int e^{x} - 3 e^{x} e^{- 2 x} d x$

Step 4.4

Multiply $e^{x}$ by $e^{- 2 x}$ by adding the exponents.

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Step 4.4.1

Move $e^{- 2 x}$ .

$\int e^{x} - 3 (e^{- 2 x} e^{x}) d x$

Step 4.4.2

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

$\int e^{x} - 3 e^{- 2 x + x} d x$

Step 4.4.3

Add $- 2 x$ and $x$ .

$\int e^{x} - 3 e^{- x} d x$

Step 5

Split the single integral into multiple integrals.

$\int e^{x} d x + \int - 3 e^{- x} d x$

Step 6

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

$e^{x} + C + \int - 3 e^{- x} d x$

Step 7

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

$e^{x} + C - 3 \int e^{- x} d x$

Step 8

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = - x$ . Find $\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate $- x$ .

$\frac{d}{d x} [- x]$

Step 8.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- \frac{d}{d x} [x]$

Step 8.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 1 \cdot 1$

Step 8.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 8.2

Rewrite the problem using $u$ and $d u$ .

$e^{x} + C - 3 \int - e^{u} d u$

Step 9

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$e^{x} + C - 3 (- \int e^{u} d u)$

Step 10

Multiply $- 1$ by $- 3$ .

$e^{x} + C + 3 \int e^{u} d u$

Step 11

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

$e^{x} + C + 3 (e^{u} + C)$

Step 12

Simplify.

$e^{x} + 3 e^{u} + C$

Step 13

Replace all occurrences of $u$ with $- x$ .

$e^{x} + 3 e^{- x} + C$

Step 14

The answer is the antiderivative of the function $f (x) = e^{x} (1 - 3 e^{- 2 x})$ .

$F (x) =$ $e^{x} + 3 e^{- x} + C$