Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 3

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

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{x^{3}}{3}$ at $10$ and at $0$ .

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

Step 6.2

Evaluate $7 x$ at $10$ and at $0$ .

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

Step 6.3

Simplify.

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Step 6.3.1

Raise $10$ to the power of $3$ .

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

Step 6.3.2

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

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

Step 6.3.3

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

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Step 6.3.3.1

Factor $3$ out of $0$ .

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

Step 6.3.3.2

Cancel the common factors.

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Step 6.3.3.2.1

Factor $3$ out of $3$ .

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

Step 6.3.3.2.2

Cancel the common factor.

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

Step 6.3.3.2.3

Rewrite the expression.

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

Step 6.3.3.2.4

Divide $0$ by $1$ .

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

Step 6.3.4

Multiply $- 1$ by $0$ .

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

Step 6.3.5

Add $\frac{1000}{3}$ and $0$ .

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

Step 6.3.6

Combine $4$ and $\frac{1000}{3}$ .

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

Step 6.3.7

Multiply $4$ by $1000$ .

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

Step 6.3.8

Multiply $7$ by $10$ .

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

Step 6.3.9

Multiply $- 7$ by $0$ .

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

Step 6.3.10

Add $70$ and $0$ .

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

Step 6.3.11

To write $70$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 6.3.12

Combine $70$ and $\frac{3}{3}$ .

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

Step 6.3.13

Combine the numerators over the common denominator.

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

Step 6.3.14

Simplify the numerator.

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Step 6.3.14.1

Multiply $70$ by $3$ .

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

Step 6.3.14.2

Add $4000$ and $210$ .

$\frac{4210}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{4210}{3}$

Decimal Form:

$1403.\overline{3}$

Mixed Number Form:

$1403 \frac{1}{3}$

Step 8