Calculus Examples

$\int_{0}^{2} (x^{2} - 1) e^{x^{3} - 3 x + 1} d x$

Step 1

Let $u = x^{3} - 3 x + 1$ . Then $d u = (3 x^{2} - 3) d x$ , so $\frac{1}{3} d u = (x^{2} - 1) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{3} - 3 x + 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{3} - 3 x + 1$ .

$\frac{d}{d x} [x^{3} - 3 x + 1]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $x^{3} - 3 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [1]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [1]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 3 x]$ .

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Step 1.1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$3 x^{2} - 3 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 1.1.3.2

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

$3 x^{2} - 3 \cdot 1 + \frac{d}{d x} [1]$

Step 1.1.3.3

Multiply $- 3$ by $1$ .

$3 x^{2} - 3 + \frac{d}{d x} [1]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$3 x^{2} - 3 + 0$

Step 1.1.4.2

Add $3 x^{2} - 3$ and $0$ .

$3 x^{2} - 3$

Step 1.2

Substitute the lower limit in for $x$ in $u = x^{3} - 3 x + 1$ .

$u_{lower} = 0^{3} - 3 \cdot 0 + 1$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 0 - 3 \cdot 0 + 1$

Step 1.3.1.2

Multiply $- 3$ by $0$ .

$u_{lower} = 0 + 0 + 1$

Step 1.3.2

Add $0$ and $0$ .

$u_{lower} = 0 + 1$

Step 1.3.3

Add $0$ and $1$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = x^{3} - 3 x + 1$ .

$u_{upper} = 2^{3} - 3 \cdot 2 + 1$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Raise $2$ to the power of $3$ .

$u_{upper} = 8 - 3 \cdot 2 + 1$

Step 1.5.1.2

Multiply $- 3$ by $2$ .

$u_{upper} = 8 - 6 + 1$

Step 1.5.2

Subtract $6$ from $8$ .

$u_{upper} = 2 + 1$

Step 1.5.3

Add $2$ and $1$ .

$u_{upper} = 3$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 3$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{1}^{3} e^{u} \frac{1}{3} d u$

Step 2

Combine $e^{u}$ and $\frac{1}{3}$ .

$\int_{1}^{3} \frac{e^{u}}{3} d u$

Step 3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int_{1}^{3} e^{u} d u$

Step 4

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

$\frac{1}{3} e^{u}]_{1}^{3}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $e^{u}$ at $3$ and at $1$ .

$\frac{1}{3} ((e^{3}) - e^{1})$

Step 5.2

Simplify.

$\frac{1}{3} (e^{3} - e)$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{1}{3} e^{3} + \frac{1}{3} (- e)$

Step 6.2

Combine $\frac{1}{3}$ and $e^{3}$ .

$\frac{e^{3}}{3} + \frac{1}{3} (- e)$

Step 6.3

Combine $\frac{1}{3}$ and $e$ .

$\frac{e^{3}}{3} - \frac{e}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{e^{3}}{3} - \frac{e}{3}$

Decimal Form:

$5.78908503 \dots$

Step 8