Calculus Examples

$\int_{0}^{1} a x + b x^{2} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{1} a x d x + \int_{0}^{1} b x^{2} d x$

Step 2

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

$a \int_{0}^{1} x d x + \int_{0}^{1} b x^{2} d x$

Step 3

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

$a \frac{1}{2} x^{2}]_{0}^{1} + \int_{0}^{1} b x^{2} d x$

Step 4

Combine $\frac{1}{2}$ and $x^{2}$ .

$a \frac{x^{2}}{2}]_{0}^{1} + \int_{0}^{1} b x^{2} d x$

Step 5

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

$a \frac{x^{2}}{2}]_{0}^{1} + b \int_{0}^{1} x^{2} d x$

Step 6

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

$a \frac{x^{2}}{2}]_{0}^{1} + b \frac{1}{3} x^{3}]_{0}^{1}$

Step 7

Simplify the answer.

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Step 7.1

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

$a \frac{x^{2}}{2}]_{0}^{1} + b \frac{x^{3}}{3}]_{0}^{1}$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{x^{2}}{2}$ at $1$ and at $0$ .

$a ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) + b \frac{x^{3}}{3}]_{0}^{1}$

Step 7.2.2

Evaluate $\frac{x^{3}}{3}$ at $1$ and at $0$ .

$a (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + b (\frac{1^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3

Simplify.

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Step 7.2.3.1

One to any power is one.

$a (\frac{1}{2} - \frac{0^{2}}{2}) + b (\frac{1^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.2

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

$a (\frac{1}{2} - \frac{0}{2}) + b (\frac{1^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.3

Cancel the common factor of $0$ and $2$ .

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Step 7.2.3.3.1

Factor $2$ out of $0$ .

$a (\frac{1}{2} - \frac{2 (0)}{2}) + b (\frac{1^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.3.2

Cancel the common factors.

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Step 7.2.3.3.2.1

Factor $2$ out of $2$ .

$a (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + b (\frac{1^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.3.2.2

Cancel the common factor.

$a (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + b (\frac{1^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.3.2.3

Rewrite the expression.

$a (\frac{1}{2} - \frac{0}{1}) + b (\frac{1^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.3.2.4

Divide $0$ by $1$ .

$a (\frac{1}{2} - 0) + b (\frac{1^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.4

Multiply $- 1$ by $0$ .

$a (\frac{1}{2} + 0) + b (\frac{1^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.5

Add $\frac{1}{2}$ and $0$ .

$a \frac{1}{2} + b (\frac{1^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.6

Combine $a$ and $\frac{1}{2}$ .

$\frac{a}{2} + b (\frac{1^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.7

One to any power is one.

$\frac{a}{2} + b (\frac{1}{3} - \frac{0^{3}}{3})$

Step 7.2.3.8

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

$\frac{a}{2} + b (\frac{1}{3} - \frac{0}{3})$

Step 7.2.3.9

Cancel the common factor of $0$ and $3$ .

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Step 7.2.3.9.1

Factor $3$ out of $0$ .

$\frac{a}{2} + b (\frac{1}{3} - \frac{3 (0)}{3})$

Step 7.2.3.9.2

Cancel the common factors.

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Step 7.2.3.9.2.1

Factor $3$ out of $3$ .

$\frac{a}{2} + b (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 7.2.3.9.2.2

Cancel the common factor.

$\frac{a}{2} + b (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 7.2.3.9.2.3

Rewrite the expression.

$\frac{a}{2} + b (\frac{1}{3} - \frac{0}{1})$

Step 7.2.3.9.2.4

Divide $0$ by $1$ .

$\frac{a}{2} + b (\frac{1}{3} - 0)$

Step 7.2.3.10

Multiply $- 1$ by $0$ .

$\frac{a}{2} + b (\frac{1}{3} + 0)$

Step 7.2.3.11

Add $\frac{1}{3}$ and $0$ .

$\frac{a}{2} + b \frac{1}{3}$

Step 7.2.3.12

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

$\frac{a}{2} + \frac{b}{3}$

Step 7.3

Reorder terms.

$\frac{1}{2} a + \frac{1}{3} b$

Step 8

Simplify each term.

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Step 8.1

Combine $\frac{1}{2}$ and $a$ .

$\frac{a}{2} + \frac{1}{3} b$

Step 8.2

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

$\frac{a}{2} + \frac{b}{3}$

Step 9