Calculus Examples

$\int_{5}^{\sqrt{x}} f (t) d t = x^{2}$

Step 1

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

$f \int_{5}^{\sqrt{x}} t d t$

Step 2

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

$f \frac{1}{2} t^{2}]_{5}^{\sqrt{x}}$

Step 3

Simplify the answer.

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Step 3.1

Combine $\frac{1}{2}$ and $t^{2}$ .

$f \frac{t^{2}}{2}]_{5}^{\sqrt{x}}$

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate $\frac{t^{2}}{2}$ at $\sqrt{x}$ and at $5$ .

$f ((\frac{{\sqrt{x}}^{2}}{2}) - \frac{5^{2}}{2})$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 3.2.2.1.1

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

$f (\frac{{(x^{\frac{1}{2}})}^{2}}{2} - \frac{5^{2}}{2})$

Step 3.2.2.1.2

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

$f (\frac{x^{\frac{1}{2} \cdot 2}}{2} - \frac{5^{2}}{2})$

Step 3.2.2.1.3

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

$f (\frac{x^{\frac{2}{2}}}{2} - \frac{5^{2}}{2})$

Step 3.2.2.1.4

Cancel the common factor of $2$ .

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Step 3.2.2.1.4.1

Cancel the common factor.

$f (\frac{x^{\frac{2}{2}}}{2} - \frac{5^{2}}{2})$

Step 3.2.2.1.4.2

Rewrite the expression.

$f (\frac{x^{1}}{2} - \frac{5^{2}}{2})$

Step 3.2.2.1.5

Simplify.

$f (\frac{x}{2} - \frac{5^{2}}{2})$

Step 3.2.2.2

Raise $5$ to the power of $2$ .

$f (\frac{x}{2} - \frac{25}{2})$

Step 3.3

Reorder terms.

$f (\frac{1}{2} x - \frac{25}{2})$

Step 4

Simplify.

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Step 4.1

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

$f (\frac{x}{2} - \frac{25}{2})$

Step 4.2

Apply the distributive property.

$f \frac{x}{2} + f (- \frac{25}{2})$

Step 4.3

Combine $f$ and $\frac{x}{2}$ .

$\frac{f x}{2} + f (- \frac{25}{2})$

Step 4.4

Combine $f$ and $\frac{25}{2}$ .

$\frac{f x}{2} - \frac{f \cdot 25}{2}$

Step 4.5

Move $25$ to the left of $f$ .

$\frac{f x}{2} - \frac{25 f}{2}$