Calculus Examples

$\int_{0}^{1} 0.5 \cdot t^{2} \cdot t^{- 0.5} d t$

Step 1

Multiply $t^{2}$ by $t^{- 0.5}$ by adding the exponents.

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Step 1.1

Move $t^{- 0.5}$ .

$\int_{0}^{1} 0.5 (t^{- 0.5} t^{2}) d t$

Step 1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{0}^{1} 0.5 t^{- 0.5 + 2} d t$

Step 1.3

Add $- 0.5$ and $2$ .

$\int_{0}^{1} 0.5 t^{1.5} d t$

Step 2

Since $0.5$ is constant with respect to $t$ , move $0.5$ out of the integral.

$0.5 \int_{0}^{1} t^{1.5} d t$

Step 3

By the Power Rule, the integral of $t^{1.5}$ with respect to $t$ is $\frac{1}{2.5} t^{2.5}$ .

$0.5 (\frac{1}{2.5} t^{2.5}]_{0}^{1})$

Step 4

Simplify the answer.

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Step 4.1

Combine $\frac{1}{2.5}$ and $t^{2.5}$ .

$0.5 (\frac{t^{2.5}}{2.5}]_{0}^{1})$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $\frac{t^{2.5}}{2.5}$ at $1$ and at $0$ .

$0.5 ((\frac{1^{2.5}}{2.5}) - \frac{0^{2.5}}{2.5})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

One to any power is one.

$0.5 (\frac{1}{2.5} - \frac{0^{2.5}}{2.5})$

Step 4.2.2.2

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

$0.5 (\frac{1}{2.5} - \frac{0}{2.5})$

Step 4.2.2.3

Combine the numerators over the common denominator.

$0.5 \frac{1 + 0}{2.5}$

Step 4.2.2.4

Add $1$ and $0$ .

$0.5 (\frac{1}{2.5})$

Step 4.2.2.5

Combine $0.5$ and $\frac{1}{2.5}$ .

$\frac{0.5}{2.5}$

Step 5

Divide $0.5$ by $2.5$ .

$0.2$

Step 6