Calculus Examples

$\int_{1}^{4} \frac{1}{\sqrt{t}} d t$

Step 1

Apply basic rules of exponents.

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Step 1.1

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

$\int_{1}^{4} \frac{1}{t^{\frac{1}{2}}} d t$

Step 1.2

Move $t^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int_{1}^{4} {(t^{\frac{1}{2}})}^{- 1} d t$

Step 1.3

Multiply the exponents in ${(t^{\frac{1}{2}})}^{- 1}$ .

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Step 1.3.1

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

$\int_{1}^{4} t^{\frac{1}{2} \cdot - 1} d t$

Step 1.3.2

Combine $\frac{1}{2}$ and $- 1$ .

$\int_{1}^{4} t^{\frac{- 1}{2}} d t$

Step 1.3.3

Move the negative in front of the fraction.

$\int_{1}^{4} t^{- \frac{1}{2}} d t$

Step 2

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

$2 t^{\frac{1}{2}}]_{1}^{4}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $2 t^{\frac{1}{2}}$ at $4$ and at $1$ .

$(2 \cdot 4^{\frac{1}{2}}) - 2 \cdot 1^{\frac{1}{2}}$

Step 3.2

Simplify.

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Step 3.2.1

Rewrite $4$ as $2^{2}$ .

$2 \cdot {(2^{2})}^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}}$

Step 3.2.2

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

$2 \cdot 2^{2 (\frac{1}{2})} - 2 \cdot 1^{\frac{1}{2}}$

Step 3.2.3

Cancel the common factor of $2$ .

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Step 3.2.3.1

Cancel the common factor.

$2 \cdot 2^{2 (\frac{1}{2})} - 2 \cdot 1^{\frac{1}{2}}$

Step 3.2.3.2

Rewrite the expression.

$2 \cdot 2^{1} - 2 \cdot 1^{\frac{1}{2}}$

Step 3.2.4

Evaluate the exponent.

$2 \cdot 2 - 2 \cdot 1^{\frac{1}{2}}$

Step 3.2.5

Multiply $2$ by $2$ .

$4 - 2 \cdot 1^{\frac{1}{2}}$

Step 3.2.6

One to any power is one.

$4 - 2 \cdot 1$

Step 3.2.7

Multiply $- 2$ by $1$ .

$4 - 2$

Step 3.2.8

Subtract $2$ from $4$ .

$2$

Step 4