Calculus Examples

$\int_{0}^{120} 10 + \sqrt{t} d t$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{120} 10 d t + \int_{0}^{120} \sqrt{t} d t$

Step 2

Apply the constant rule.

$10 t]_{0}^{120} + \int_{0}^{120} \sqrt{t} d t$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$10 t]_{0}^{120} + \int_{0}^{120} t^{\frac{1}{2}} d t$

Step 4

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

$10 t]_{0}^{120} + \frac{2}{3} t^{\frac{3}{2}}]_{0}^{120}$

Step 5

Simplify the answer.

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Step 5.1

Combine $10 t]_{0}^{120}$ and $\frac{2}{3} t^{\frac{3}{2}}]_{0}^{120}$ .

$10 t + \frac{2}{3} t^{\frac{3}{2}}]_{0}^{120}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $10 t + \frac{2}{3} t^{\frac{3}{2}}$ at $120$ and at $0$ .

$(10 \cdot 120 + \frac{2}{3} \cdot 120^{\frac{3}{2}}) - (10 \cdot 0 + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 5.2.2

Simplify.

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Step 5.2.2.1

Multiply $10$ by $120$ .

$1200 + \frac{2}{3} \cdot 120^{\frac{3}{2}} - (10 \cdot 0 + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 5.2.2.2

Combine $\frac{2}{3}$ and $120^{\frac{3}{2}}$ .

$1200 + \frac{2 \cdot 120^{\frac{3}{2}}}{3} - (10 \cdot 0 + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 5.2.2.3

Multiply $10$ by $0$ .

$1200 + \frac{2 \cdot 120^{\frac{3}{2}}}{3} - (0 + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 5.2.2.4

Rewrite $0$ as $0^{2}$ .

$1200 + \frac{2 \cdot 120^{\frac{3}{2}}}{3} - (0 + \frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}})$

Step 5.2.2.5

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$1200 + \frac{2 \cdot 120^{\frac{3}{2}}}{3} - (0 + \frac{2}{3} \cdot 0^{2 (\frac{3}{2})})$

Step 5.2.2.6

Cancel the common factor of $2$ .

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Step 5.2.2.6.1

Cancel the common factor.

$1200 + \frac{2 \cdot 120^{\frac{3}{2}}}{3} - (0 + \frac{2}{3} \cdot 0^{2 (\frac{3}{2})})$

Step 5.2.2.6.2

Rewrite the expression.

$1200 + \frac{2 \cdot 120^{\frac{3}{2}}}{3} - (0 + \frac{2}{3} \cdot 0^{3})$

Step 5.2.2.7

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

$1200 + \frac{2 \cdot 120^{\frac{3}{2}}}{3} - (0 + \frac{2}{3} \cdot 0)$

Step 5.2.2.8

Multiply $\frac{2}{3}$ by $0$ .

$1200 + \frac{2 \cdot 120^{\frac{3}{2}}}{3} - (0 + 0)$

Step 5.2.2.9

Add $0$ and $0$ .

$1200 + \frac{2 \cdot 120^{\frac{3}{2}}}{3} - 0$

Step 5.2.2.10

Multiply $- 1$ by $0$ .

$1200 + \frac{2 \cdot 120^{\frac{3}{2}}}{3} + 0$

Step 5.2.2.11

Add $1200 + \frac{2 \cdot 120^{\frac{3}{2}}}{3}$ and $0$ .

$1200 + \frac{2 \cdot 120^{\frac{3}{2}}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$1200 + \frac{2 \cdot 120^{\frac{3}{2}}}{3}$

Decimal Form:

$2076.35609200 \dots$

Step 7