Calculus Examples

$\int_{3}^{9} 3 x^{2} + 2 x - 1 d x$

Since integration is linear, the integral of $3 x^{2} + 2 x - 1$ with respect to $x$ is $\int_{3}^{9} 3 x^{2} d x + \int_{3}^{9} 2 x d x + \int_{3}^{9} - 1 d x$ .

$\int_{3}^{9} 3 x^{2} d x + \int_{3}^{9} 2 x d x + \int_{3}^{9} - 1 d x$

Since $3$ is constant with respect to $x$ , the integral of $3 x^{2}$ with respect to $x$ is $3 \int_{3}^{9} x^{2} d x$ .

$3 \int_{3}^{9} x^{2} d x + \int_{3}^{9} 2 x d x + \int_{3}^{9} - 1 d x$

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

$3 (\frac{1}{3} x^{3}]_{3}^{9}) + \int_{3}^{9} 2 x d x + \int_{3}^{9} - 1 d x$

Combine fractions.

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Write $x^{3}$ as a fraction with denominator $1$ .

$3 (\frac{1}{3} \frac{x^{3}}{1}]_{3}^{9}) + \int_{3}^{9} 2 x d x + \int_{3}^{9} - 1 d x$

Multiply $\frac{1}{3}$ and $\frac{x^{3}}{1}$ to get $\frac{x^{3}}{3}$ .

$3 (\frac{x^{3}}{3}]_{3}^{9}) + \int_{3}^{9} 2 x d x + \int_{3}^{9} - 1 d x$

Since $2$ is constant with respect to $x$ , the integral of $2 x$ with respect to $x$ is $2 \int_{3}^{9} x d x$ .

$3 (\frac{x^{3}}{3}]_{3}^{9}) + 2 \int_{3}^{9} x d x + \int_{3}^{9} - 1 d x$

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

$3 (\frac{x^{3}}{3}]_{3}^{9}) + 2 (\frac{1}{2} x^{2}]_{3}^{9}) + \int_{3}^{9} - 1 d x$

Combine fractions.

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Write $x^{2}$ as a fraction with denominator $1$ .

$3 (\frac{x^{3}}{3}]_{3}^{9}) + 2 (\frac{1}{2} \frac{x^{2}}{1}]_{3}^{9}) + \int_{3}^{9} - 1 d x$

Multiply $\frac{1}{2}$ and $\frac{x^{2}}{1}$ to get $\frac{x^{2}}{2}$ .

$3 (\frac{x^{3}}{3}]_{3}^{9}) + 2 (\frac{x^{2}}{2}]_{3}^{9}) + \int_{3}^{9} - 1 d x$

Since $- 1$ is constant with respect to $x$ , the integral of $- 1$ with respect to $x$ is $- x$ .

$3 (\frac{x^{3}}{3}]_{3}^{9}) + 2 (\frac{x^{2}}{2}]_{3}^{9}) + - x]_{3}^{9}$

Simplify the answer.

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Evaluate $\frac{x^{3}}{3}$ at $9$ and at $3$ .

$3 ((\frac{9^{3}}{3}) - \frac{3^{3}}{3}) + 2 (\frac{x^{2}}{2}]_{3}^{9}) + - x]_{3}^{9}$

Evaluate $\frac{x^{2}}{2}$ at $9$ and at $3$ .

$3 (\frac{9^{3}}{3} - \frac{3^{3}}{3}) + 2 (\frac{9^{2}}{2} - \frac{3^{2}}{2}) + - x]_{3}^{9}$

Evaluate $- x$ at $9$ and at $3$ .

$3 (\frac{9^{3}}{3} - \frac{3^{3}}{3}) + 2 (\frac{9^{2}}{2} - \frac{3^{2}}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Raise $9$ to the power of $3$ to get $729$ .

$3 (\frac{729}{3} - \frac{3^{3}}{3}) + 2 (\frac{9^{2}}{2} - \frac{3^{2}}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Reduce the expression by cancelling the common factors.

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Factor $3$ out of $3$ .

$3 (\frac{3 \cdot 243}{3 (1)} - \frac{3^{3}}{3}) + 2 (\frac{9^{2}}{2} - \frac{3^{2}}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Cancel the common factor.

$3 (\frac{3 \cdot 243}{3 \cdot 1} - \frac{3^{3}}{3}) + 2 (\frac{9^{2}}{2} - \frac{3^{2}}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Rewrite the expression.

$3 (\frac{243}{1} - \frac{3^{3}}{3}) + 2 (\frac{9^{2}}{2} - \frac{3^{2}}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Divide $243$ by $1$ to get $243$ .

$3 (243 - \frac{3^{3}}{3}) + 2 (\frac{9^{2}}{2} - \frac{3^{2}}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Raise $3$ to the power of $3$ to get $27$ .

$3 (243 - \frac{27}{3}) + 2 (\frac{9^{2}}{2} - \frac{3^{2}}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Reduce the expression by cancelling the common factors.

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Factor $3$ out of $3$ .

$3 (243 - \frac{3 \cdot 9}{3 (1)}) + 2 (\frac{9^{2}}{2} - \frac{3^{2}}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Cancel the common factor.

$3 (243 - \frac{3 \cdot 9}{3 \cdot 1}) + 2 (\frac{9^{2}}{2} - \frac{3^{2}}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Rewrite the expression.

$3 (243 - \frac{9}{1}) + 2 (\frac{9^{2}}{2} - \frac{3^{2}}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Divide $9$ by $1$ to get $9$ .

$3 (243 - 1 \cdot 9) + 2 (\frac{9^{2}}{2} - \frac{3^{2}}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Multiply $- 1$ by $9$ to get $- 9$ .

$3 (243 - 9) + 2 (\frac{9^{2}}{2} - \frac{3^{2}}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Subtract $9$ from $243$ to get $234$ .

$3 \cdot 234 + 2 (\frac{9^{2}}{2} - \frac{3^{2}}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Multiply $3$ by $234$ to get $702$ .

$702 + 2 (\frac{9^{2}}{2} - \frac{3^{2}}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Raise $9$ to the power of $2$ to get $81$ .

$702 + 2 (\frac{81}{2} - \frac{3^{2}}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Raise $3$ to the power of $2$ to get $9$ .

$702 + 2 (\frac{81}{2} - \frac{9}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Combine the numerators over the common denominator.

$702 + 2 \frac{81 - 1 \cdot 9}{2} + ((- 1 \cdot 9) + 1 \cdot 3)$

Simplify the numerator.

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Multiply $- 1$ by $9$ to get $- 9$ .

$702 + 2 \frac{81 - 9}{2} + ((- 1 \cdot 9) + 1 \cdot 3)$

Subtract $9$ from $81$ .

$702 + 2 (\frac{72}{2}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Reduce the expression by cancelling the common factors.

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Factor $2$ out of $2$ .

$702 + 2 \frac{2 \cdot 36}{2 (1)} + ((- 1 \cdot 9) + 1 \cdot 3)$

Cancel the common factor.

$702 + 2 \frac{2 \cdot 36}{2 \cdot 1} + ((- 1 \cdot 9) + 1 \cdot 3)$

Rewrite the expression.

$702 + 2 (\frac{36}{1}) + ((- 1 \cdot 9) + 1 \cdot 3)$

Divide $36$ by $1$ to get $36$ .

$702 + 2 \cdot 36 + ((- 1 \cdot 9) + 1 \cdot 3)$

Multiply $2$ by $36$ to get $72$ .

$702 + 72 + ((- 1 \cdot 9) + 1 \cdot 3)$

Add $702$ and $72$ to get $774$ .

$774 + ((- 1 \cdot 9) + 1 \cdot 3)$

Multiply $- 1$ by $9$ to get $- 9$ .

$774 + (- 9 + 1 \cdot 3)$

Multiply $3$ by $1$ to get $3$ .

$774 + (- 9 + 3)$

Add $- 9$ and $3$ to get $- 6$ .

$774 - 6$

Subtract $6$ from $774$ to get $768$ .

$768$

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