Calculus Examples

$\int_{2}^{x} (3 t^{2} - 1) d t$

Step 1

Remove parentheses.

$\int_{2}^{x} 3 t^{2} - 1 d t$

Step 2

Split the single integral into multiple integrals.

$\int_{2}^{x} 3 t^{2} d t + \int_{2}^{x} - 1 d t$

Step 3

Since $3$ is constant with respect to $t$ , move $3$ out of the integral.

$3 \int_{2}^{x} t^{2} d t + \int_{2}^{x} - 1 d t$

Step 4

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

$3 (\frac{1}{3} t^{3}]_{2}^{x}) + \int_{2}^{x} - 1 d t$

Step 5

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

$3 (\frac{t^{3}}{3}]_{2}^{x}) + \int_{2}^{x} - 1 d t$

Step 6

Apply the constant rule.

$3 (\frac{t^{3}}{3}]_{2}^{x}) + - t]_{2}^{x}$

Step 7

Simplify the answer.

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Step 7.1

Substitute and simplify.

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Step 7.1.1

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

$3 ((\frac{x^{3}}{3}) - \frac{2^{3}}{3}) + - t]_{2}^{x}$

Step 7.1.2

Evaluate $- t$ at $x$ and at $2$ .

$3 (\frac{x^{3}}{3} - \frac{2^{3}}{3}) + - x + 1 \cdot 2$

Step 7.1.3

Simplify.

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Step 7.1.3.1

Raise $2$ to the power of $3$ .

$3 (\frac{x^{3}}{3} - \frac{8}{3}) - x + 1 \cdot 2$

Step 7.1.3.2

Multiply $2$ by $1$ .

$3 (\frac{x^{3}}{3} - \frac{8}{3}) - x + 2$

Step 7.2

Reorder terms.

$3 (\frac{1}{3} x^{3} - \frac{8}{3}) - x + 2$

Step 8

Simplify.

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Step 8.1

Simplify each term.

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Step 8.1.1

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

$3 (\frac{x^{3}}{3} - \frac{8}{3}) - x + 2$

Step 8.1.2

Apply the distributive property.

$3 \frac{x^{3}}{3} + 3 (- \frac{8}{3}) - x + 2$

Step 8.1.3

Cancel the common factor of $3$ .

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Step 8.1.3.1

Cancel the common factor.

$3 \frac{x^{3}}{3} + 3 (- \frac{8}{3}) - x + 2$

Step 8.1.3.2

Rewrite the expression.

$x^{3} + 3 (- \frac{8}{3}) - x + 2$

Step 8.1.4

Cancel the common factor of $3$ .

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Step 8.1.4.1

Move the leading negative in $- \frac{8}{3}$ into the numerator.

$x^{3} + 3 (\frac{- 8}{3}) - x + 2$

Step 8.1.4.2

Cancel the common factor.

$x^{3} + 3 (\frac{- 8}{3}) - x + 2$

Step 8.1.4.3

Rewrite the expression.

$x^{3} - 8 - x + 2$

Step 8.2

Add $- 8$ and $2$ .

$x^{3} - x - 6$