Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Apply the constant rule.

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

Step 7

Simplify the answer.

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Step 7.1

Substitute and simplify.

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Step 7.1.1

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

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

Step 7.1.2

Evaluate $b x$ at $1$ and at $0$ .

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

Step 7.1.3

Simplify.

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Step 7.1.3.1

One to any power is one.

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

Step 7.1.3.2

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

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

Step 7.1.3.3

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

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Step 7.1.3.3.1

Factor $3$ out of $0$ .

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

Step 7.1.3.3.2

Cancel the common factors.

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Step 7.1.3.3.2.1

Factor $3$ out of $3$ .

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

Step 7.1.3.3.2.2

Cancel the common factor.

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

Step 7.1.3.3.2.3

Rewrite the expression.

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

Step 7.1.3.3.2.4

Divide $0$ by $1$ .

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

Step 7.1.3.4

Multiply $- 1$ by $0$ .

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

Step 7.1.3.5

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

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

Step 7.1.3.6

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

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

Step 7.1.3.7

Multiply $b$ by $1$ .

$\frac{a}{3} + b - b \cdot 0$

Step 7.1.3.8

Multiply $0$ by $- 1$ .

$\frac{a}{3} + b + 0 b$

Step 7.1.3.9

Multiply $0$ by $b$ .

$\frac{a}{3} + b + 0$

Step 7.1.3.10

Add $b$ and $0$ .

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

Step 7.2

Reorder terms.

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

Step 8

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

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

Step 9