Calculus Examples

$\int_{0}^{4} {(x - 3)}^{2} + 3 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{4} {(x - 3)}^{2} d x + \int_{0}^{4} 3 d x$

Step 2

Let $u = x - 3$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

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

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Step 2.1.1

Differentiate $x - 3$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.1.5

Add $1$ and $0$ .

$1$

Step 2.2

Substitute the lower limit in for $x$ in $u = x - 3$ .

$u_{lower} = 0 - 3$

Step 2.3

Subtract $3$ from $0$ .

$u_{lower} = - 3$

Step 2.4

Substitute the upper limit in for $x$ in $u = x - 3$ .

$u_{upper} = 4 - 3$

Step 2.5

Subtract $3$ from $4$ .

$u_{upper} = 1$

Step 2.6

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

$u_{lower} = - 3$

$u_{upper} = 1$

Step 2.7

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

$\int_{- 3}^{1} u^{2} d u + \int_{0}^{4} 3 d x$

Step 3

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$\frac{1}{3} u^{3}]_{- 3}^{1} + \int_{0}^{4} 3 d x$

Step 4

Apply the constant rule.

$\frac{1}{3} u^{3}]_{- 3}^{1} + 3 x]_{0}^{4}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{1}{3} u^{3}$ at $1$ and at $- 3$ .

$(\frac{1}{3} \cdot 1^{3}) - \frac{1}{3} {(- 3)}^{3} + 3 x]_{0}^{4}$

Step 5.2

Evaluate $3 x$ at $4$ and at $0$ .

$(\frac{1}{3} \cdot 1^{3}) - \frac{1}{3} {(- 3)}^{3} + (3 \cdot 4) - 3 \cdot 0$

Step 5.3

Simplify.

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Step 5.3.1

One to any power is one.

$\frac{1}{3} \cdot 1 - \frac{1}{3} {(- 3)}^{3} + (3 \cdot 4) - 3 \cdot 0$

Step 5.3.2

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3} - \frac{1}{3} {(- 3)}^{3} + (3 \cdot 4) - 3 \cdot 0$

Step 5.3.3

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

$\frac{1}{3} - \frac{1}{3} \cdot - 27 + (3 \cdot 4) - 3 \cdot 0$

Step 5.3.4

Multiply $- 27$ by $- 1$ .

$\frac{1}{3} + 27 (\frac{1}{3}) + (3 \cdot 4) - 3 \cdot 0$

Step 5.3.5

Combine $27$ and $\frac{1}{3}$ .

$\frac{1}{3} + \frac{27}{3} + (3 \cdot 4) - 3 \cdot 0$

Step 5.3.6

Cancel the common factor of $27$ and $3$ .

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Step 5.3.6.1

Factor $3$ out of $27$ .

$\frac{1}{3} + \frac{3 \cdot 9}{3} + (3 \cdot 4) - 3 \cdot 0$

Step 5.3.6.2

Cancel the common factors.

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Step 5.3.6.2.1

Factor $3$ out of $3$ .

$\frac{1}{3} + \frac{3 \cdot 9}{3 (1)} + (3 \cdot 4) - 3 \cdot 0$

Step 5.3.6.2.2

Cancel the common factor.

$\frac{1}{3} + \frac{3 \cdot 9}{3 \cdot 1} + (3 \cdot 4) - 3 \cdot 0$

Step 5.3.6.2.3

Rewrite the expression.

$\frac{1}{3} + \frac{9}{1} + (3 \cdot 4) - 3 \cdot 0$

Step 5.3.6.2.4

Divide $9$ by $1$ .

$\frac{1}{3} + 9 + (3 \cdot 4) - 3 \cdot 0$

Step 5.3.7

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3} + 9 \cdot \frac{3}{3} + (3 \cdot 4) - 3 \cdot 0$

Step 5.3.8

Combine $9$ and $\frac{3}{3}$ .

$\frac{1}{3} + \frac{9 \cdot 3}{3} + (3 \cdot 4) - 3 \cdot 0$

Step 5.3.9

Combine the numerators over the common denominator.

$\frac{1 + 9 \cdot 3}{3} + (3 \cdot 4) - 3 \cdot 0$

Step 5.3.10

Simplify the numerator.

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Step 5.3.10.1

Multiply $9$ by $3$ .

$\frac{1 + 27}{3} + (3 \cdot 4) - 3 \cdot 0$

Step 5.3.10.2

Add $1$ and $27$ .

$\frac{28}{3} + (3 \cdot 4) - 3 \cdot 0$

Step 5.3.11

Multiply $3$ by $4$ .

$\frac{28}{3} + 12 - 3 \cdot 0$

Step 5.3.12

Multiply $- 3$ by $0$ .

$\frac{28}{3} + 12 + 0$

Step 5.3.13

Add $12$ and $0$ .

$\frac{28}{3} + 12$

Step 5.3.14

To write $12$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{28}{3} + 12 \cdot \frac{3}{3}$

Step 5.3.15

Combine $12$ and $\frac{3}{3}$ .

$\frac{28}{3} + \frac{12 \cdot 3}{3}$

Step 5.3.16

Combine the numerators over the common denominator.

$\frac{28 + 12 \cdot 3}{3}$

Step 5.3.17

Simplify the numerator.

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Step 5.3.17.1

Multiply $12$ by $3$ .

$\frac{28 + 36}{3}$

Step 5.3.17.2

Add $28$ and $36$ .

$\frac{64}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{64}{3}$

Decimal Form:

$21.\overline{3}$

Mixed Number Form:

$21 \frac{1}{3}$

Step 7