Calculus Examples

$\int_{0}^{x^{3}} f (t) d t = x^{4}$

Step 1

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

$f \int_{0}^{x^{3}} t d t$

Step 2

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

$f \frac{1}{2} t^{2}]_{0}^{x^{3}}$

Step 3

Simplify the answer.

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Step 3.1

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

$f \frac{t^{2}}{2}]_{0}^{x^{3}}$

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate $\frac{t^{2}}{2}$ at $x^{3}$ and at $0$ .

$f ((\frac{{(x^{3})}^{2}}{2}) - \frac{0^{2}}{2})$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 3.2.2.1.1

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

$f (\frac{x^{3 \cdot 2}}{2} - \frac{0^{2}}{2})$

Step 3.2.2.1.2

Multiply $3$ by $2$ .

$f (\frac{x^{6}}{2} - \frac{0^{2}}{2})$

Step 3.2.2.2

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

$f (\frac{x^{6}}{2} - \frac{0}{2})$

Step 3.2.2.3

Cancel the common factor of $0$ and $2$ .

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Step 3.2.2.3.1

Factor $2$ out of $0$ .

$f (\frac{x^{6}}{2} - \frac{2 (0)}{2})$

Step 3.2.2.3.2

Cancel the common factors.

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Step 3.2.2.3.2.1

Factor $2$ out of $2$ .

$f (\frac{x^{6}}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 3.2.2.3.2.2

Cancel the common factor.

$f (\frac{x^{6}}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 3.2.2.3.2.3

Rewrite the expression.

$f (\frac{x^{6}}{2} - \frac{0}{1})$

Step 3.2.2.3.2.4

Divide $0$ by $1$ .

$f (\frac{x^{6}}{2} - 0)$

Step 3.2.2.4

Multiply $- 1$ by $0$ .

$f (\frac{x^{6}}{2} + 0)$

Step 3.2.2.5

Add $\frac{x^{6}}{2}$ and $0$ .

$f \frac{x^{6}}{2}$

Step 3.2.2.6

Combine $f$ and $\frac{x^{6}}{2}$ .

$\frac{f x^{6}}{2}$

$\frac{1}{2} f x^{6}$

Step 4

Simplify.

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Step 4.1

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

$\frac{f}{2} x^{6}$

Step 4.2

Combine $\frac{f}{2}$ and $x^{6}$ .

$\frac{f x^{6}}{2}$