Calculus Examples

$e^{x} (- x^{3} + 3 x) - 3$

Step 1

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

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

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.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = - x^{3} + 3 x$ and $d v = e^{x}$ .

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

Step 6

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = - 3 x^{2} + 3$ and $d v = e^{x}$ .

$(- x^{3} + 3 x) e^{x} - ((- 3 x^{2} + 3) e^{x} - \int e^{x} (- 6 x) d x) + \int - 3 d x$

Step 7

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$(- x^{3} + 3 x) e^{x} - ((- 3 x^{2} + 3) e^{x} - (- 6 \int e^{x} (x) d x)) + \int - 3 d x$

Step 8

Multiply $- 6$ by $- 1$ .

$(- x^{3} + 3 x) e^{x} - ((- 3 x^{2} + 3) e^{x} + 6 \int e^{x} (x) d x) + \int - 3 d x$

Step 9

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{x}$ .

$(- x^{3} + 3 x) e^{x} - ((- 3 x^{2} + 3) e^{x} + 6 (x e^{x} - \int e^{x} d x)) + \int - 3 d x$

Step 10

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

$(- x^{3} + 3 x) e^{x} - ((- 3 x^{2} + 3) e^{x} + 6 (x e^{x} - (e^{x} + C))) + \int - 3 d x$

Step 11

Apply the constant rule.

$(- x^{3} + 3 x) e^{x} - ((- 3 x^{2} + 3) e^{x} + 6 (x e^{x} - (e^{x} + C))) - 3 x + C$

Step 12

Simplify.

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Step 12.1

Simplify.

$- x^{3} e^{x} + 3 x e^{x} + 3 x^{2} e^{x} - 6 x e^{x} + 3 e^{x} - 3 x + C$

Step 12.2

Subtract $6 x e^{x}$ from $3 x e^{x}$ .

$- x^{3} e^{x} - 3 x e^{x} + 3 x^{2} e^{x} + 3 e^{x} - 3 x + C$

Step 13

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

$F (x) =$ $- x^{3} e^{x} - 3 x e^{x} + 3 x^{2} e^{x} + 3 e^{x} - 3 x + C$