Calculus Examples

\int_{0}^{10} 4 x^{2} + 7 d x

$\int_{0}^{10} 4 x^{2} + 7 d x$

Step 1

Split the single integral into multiple integrals.

\int_{0}^{10} 4 x^{2} d x + \int_{0}^{10} 7 d x

$\int_{0}^{10} 4 x^{2} d x + \int_{0}^{10} 7 d x$

Step 2

Since

4

$4$ is constant with respect to

x

$x$ , move

4

$4$ out of the integral.

4 \int_{0}^{10} x^{2} d x + \int_{0}^{10} 7 d x

$4 \int_{0}^{10} x^{2} d x + \int_{0}^{10} 7 d x$

Step 3

By the Power Rule, the integral of

x^{2}

$x^{2}$ with respect to

x

$x$ is

\frac{1}{3} x^{3}

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

4 (\frac{1}{3} x^{3}]_{0}^{10}) + \int_{0}^{10} 7 d x

$4 (\frac{1}{3} x^{3}]_{0}^{10}) + \int_{0}^{10} 7 d x$

Step 4

Combine

\frac{1}{3}

$\frac{1}{3}$ and

x^{3}

$x^{3}$ .

4 (\frac{x^{3}}{3}]_{0}^{10}) + \int_{0}^{10} 7 d x

$4 (\frac{x^{3}}{3}]_{0}^{10}) + \int_{0}^{10} 7 d x$

Step 5

Apply the constant rule.

4 (\frac{x^{3}}{3}]_{0}^{10}) + 7 x]_{0}^{10}

$4 (\frac{x^{3}}{3}]_{0}^{10}) + 7 x]_{0}^{10}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate

\frac{x^{3}}{3}

$\frac{x^{3}}{3}$ at

10

$10$ and at

0

$0$ .

4 ((\frac{10^{3}}{3}) - \frac{0^{3}}{3}) + 7 x]_{0}^{10}

$4 ((\frac{10^{3}}{3}) - \frac{0^{3}}{3}) + 7 x]_{0}^{10}$

Step 6.2

Evaluate

7 x

$7 x$ at

10

$10$ and at

0

$0$ .

4 (\frac{10^{3}}{3} - \frac{0^{3}}{3}) + 7 \cdot 10 - 7 \cdot 0

$4 (\frac{10^{3}}{3} - \frac{0^{3}}{3}) + 7 \cdot 10 - 7 \cdot 0$

Step 6.3

Simplify.

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Step 6.3.1

Raise

10

$10$ to the power of

3

$3$ .

4 (\frac{1000}{3} - \frac{0^{3}}{3}) + 7 \cdot 10 - 7 \cdot 0

$4 (\frac{1000}{3} - \frac{0^{3}}{3}) + 7 \cdot 10 - 7 \cdot 0$

Step 6.3.2

Raising

0

$0$ to any positive power yields

0

$0$ .

4 (\frac{1000}{3} - \frac{0}{3}) + 7 \cdot 10 - 7 \cdot 0

$4 (\frac{1000}{3} - \frac{0}{3}) + 7 \cdot 10 - 7 \cdot 0$

Step 6.3.3

Cancel the common factor of

0

$0$ and

3

$3$ .

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Step 6.3.3.1

Factor

3

$3$ out of

0

$0$ .

4 (\frac{1000}{3} - \frac{3 (0)}{3}) + 7 \cdot 10 - 7 \cdot 0

$4 (\frac{1000}{3} - \frac{3 (0)}{3}) + 7 \cdot 10 - 7 \cdot 0$

Step 6.3.3.2

Cancel the common factors.

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Step 6.3.3.2.1

Factor

3

$3$ out of

3

$3$ .

4 (\frac{1000}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 7 \cdot 10 - 7 \cdot 0

$4 (\frac{1000}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 7 \cdot 10 - 7 \cdot 0$

Step 6.3.3.2.2

Cancel the common factor.

4 (\frac{1000}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 7 \cdot 10 - 7 \cdot 0

$4 (\frac{1000}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 7 \cdot 10 - 7 \cdot 0$

Step 6.3.3.2.3

Rewrite the expression.

4 (\frac{1000}{3} - \frac{0}{1}) + 7 \cdot 10 - 7 \cdot 0

$4 (\frac{1000}{3} - \frac{0}{1}) + 7 \cdot 10 - 7 \cdot 0$

Step 6.3.3.2.4

Divide

0

$0$ by

1

$1$ .

4 (\frac{1000}{3} - 0) + 7 \cdot 10 - 7 \cdot 0

$4 (\frac{1000}{3} - 0) + 7 \cdot 10 - 7 \cdot 0$

4 (\frac{1000}{3} - 0) + 7 \cdot 10 - 7 \cdot 0

$4 (\frac{1000}{3} - 0) + 7 \cdot 10 - 7 \cdot 0$

4 (\frac{1000}{3} - 0) + 7 \cdot 10 - 7 \cdot 0

$4 (\frac{1000}{3} - 0) + 7 \cdot 10 - 7 \cdot 0$

Step 6.3.4

Multiply

- 1

$- 1$ by

0

$0$ .

4 (\frac{1000}{3} + 0) + 7 \cdot 10 - 7 \cdot 0

$4 (\frac{1000}{3} + 0) + 7 \cdot 10 - 7 \cdot 0$

Step 6.3.5

Add

\frac{1000}{3}

$\frac{1000}{3}$ and

0

$0$ .

4 (\frac{1000}{3}) + 7 \cdot 10 - 7 \cdot 0

$4 (\frac{1000}{3}) + 7 \cdot 10 - 7 \cdot 0$

Step 6.3.6

Combine

4

$4$ and

\frac{1000}{3}

$\frac{1000}{3}$ .

\frac{4 \cdot 1000}{3} + 7 \cdot 10 - 7 \cdot 0

$\frac{4 \cdot 1000}{3} + 7 \cdot 10 - 7 \cdot 0$

Step 6.3.7

Multiply

4

$4$ by

1000

$1000$ .

\frac{4000}{3} + 7 \cdot 10 - 7 \cdot 0

$\frac{4000}{3} + 7 \cdot 10 - 7 \cdot 0$

Step 6.3.8

Multiply

7

$7$ by

10

$10$ .

\frac{4000}{3} + 70 - 7 \cdot 0

$\frac{4000}{3} + 70 - 7 \cdot 0$

Step 6.3.9

Multiply

- 7

$- 7$ by

0

$0$ .

\frac{4000}{3} + 70 + 0

$\frac{4000}{3} + 70 + 0$

Step 6.3.10

Add

70

$70$ and

0

$0$ .

\frac{4000}{3} + 70

$\frac{4000}{3} + 70$

Step 6.3.11

To write

70

$70$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

\frac{4000}{3} + 70 \cdot \frac{3}{3}

$\frac{4000}{3} + 70 \cdot \frac{3}{3}$

Step 6.3.12

Combine

70

$70$ and

\frac{3}{3}

$\frac{3}{3}$ .

\frac{4000}{3} + \frac{70 \cdot 3}{3}

$\frac{4000}{3} + \frac{70 \cdot 3}{3}$

Step 6.3.13

Combine the numerators over the common denominator.

\frac{4000 + 70 \cdot 3}{3}

$\frac{4000 + 70 \cdot 3}{3}$

Step 6.3.14

Simplify the numerator.

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Step 6.3.14.1

Multiply

70

$70$ by

3

$3$ .

\frac{4000 + 210}{3}

$\frac{4000 + 210}{3}$

Step 6.3.14.2

Add

4000

$4000$ and

210

$210$ .

\frac{4210}{3}

$\frac{4210}{3}$

\frac{4210}{3}

$\frac{4210}{3}$

\frac{4210}{3}

$\frac{4210}{3}$

\frac{4210}{3}

$\frac{4210}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

\frac{4210}{3}

$\frac{4210}{3}$

Decimal Form:

1403.\overline{3}

$1403.\overline{3}$

Mixed Number Form:

1403 \frac{1}{3}

$1403 \frac{1}{3}$

Step 8