Calculus Examples

$\int_{x}^{1} e t^{2} d t$

Step 1

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

$e \int_{x}^{1} t^{2} d t$

Step 2

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

$e \frac{1}{3} t^{3}]_{x}^{1}$

Step 3

Simplify the answer.

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Step 3.1

Combine $\frac{1}{3}$ and $t^{3}$ .

$e \frac{t^{3}}{3}]_{x}^{1}$

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate $\frac{t^{3}}{3}$ at $1$ and at $x$ .

$e ((\frac{1^{3}}{3}) - \frac{x^{3}}{3})$

Step 3.2.2

One to any power is one.

$e (\frac{1}{3} - \frac{x^{3}}{3})$

Step 3.3

Reorder terms.

$e (\frac{1}{3} - \frac{1}{3} x^{3})$

Step 4

Simplify.

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Step 4.1

Combine $x^{3}$ and $\frac{1}{3}$ .

$e (\frac{1}{3} - \frac{x^{3}}{3})$

Step 4.2

Apply the distributive property.

$e \frac{1}{3} + e (- \frac{x^{3}}{3})$

Step 4.3

Combine $e$ and $\frac{1}{3}$ .

$\frac{e}{3} + e (- \frac{x^{3}}{3})$

Step 4.4

Combine $e$ and $\frac{x^{3}}{3}$ .

$\frac{e}{3} - \frac{e x^{3}}{3}$