Calculus Examples

$\int_{3}^{5} 3 x^{2} + 2 x - 1 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{3}^{5} 3 x^{2} d x + \int_{3}^{5} 2 x d x + \int_{3}^{5} - 1 d x$

Step 2

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

$3 \int_{3}^{5} x^{2} d x + \int_{3}^{5} 2 x d x + \int_{3}^{5} - 1 d x$

Step 3

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

$3 (\frac{1}{3} x^{3}]_{3}^{5}) + \int_{3}^{5} 2 x d x + \int_{3}^{5} - 1 d x$

Step 4

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

$3 (\frac{x^{3}}{3}]_{3}^{5}) + \int_{3}^{5} 2 x d x + \int_{3}^{5} - 1 d x$

Step 5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$3 (\frac{x^{3}}{3}]_{3}^{5}) + 2 \int_{3}^{5} x d x + \int_{3}^{5} - 1 d x$

Step 6

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

$3 (\frac{x^{3}}{3}]_{3}^{5}) + 2 (\frac{1}{2} x^{2}]_{3}^{5}) + \int_{3}^{5} - 1 d x$

Step 7

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

$3 (\frac{x^{3}}{3}]_{3}^{5}) + 2 (\frac{x^{2}}{2}]_{3}^{5}) + \int_{3}^{5} - 1 d x$

Step 8

Apply the constant rule.

$3 (\frac{x^{3}}{3}]_{3}^{5}) + 2 (\frac{x^{2}}{2}]_{3}^{5}) + - x]_{3}^{5}$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $\frac{x^{3}}{3}$ at $5$ and at $3$ .

$3 ((\frac{5^{3}}{3}) - \frac{3^{3}}{3}) + 2 (\frac{x^{2}}{2}]_{3}^{5}) + - x]_{3}^{5}$

Step 9.2

Evaluate $\frac{x^{2}}{2}$ at $5$ and at $3$ .

$3 (\frac{5^{3}}{3} - \frac{3^{3}}{3}) + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + - x]_{3}^{5}$

Step 9.3

Evaluate $- x$ at $5$ and at $3$ .

$3 (\frac{5^{3}}{3} - \frac{3^{3}}{3}) + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4

Simplify.

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Step 9.4.1

Raise $5$ to the power of $3$ .

$3 (\frac{125}{3} - \frac{3^{3}}{3}) + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.2

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

$3 (\frac{125}{3} - \frac{27}{3}) + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.3

Cancel the common factor of $27$ and $3$ .

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Step 9.4.3.1

Factor $3$ out of $27$ .

$3 (\frac{125}{3} - \frac{3 \cdot 9}{3}) + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.3.2

Cancel the common factors.

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Step 9.4.3.2.1

Factor $3$ out of $3$ .

$3 (\frac{125}{3} - \frac{3 \cdot 9}{3 (1)}) + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.3.2.2

Cancel the common factor.

$3 (\frac{125}{3} - \frac{3 \cdot 9}{3 \cdot 1}) + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.3.2.3

Rewrite the expression.

$3 (\frac{125}{3} - \frac{9}{1}) + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.3.2.4

Divide $9$ by $1$ .

$3 (\frac{125}{3} - 1 \cdot 9) + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.4

Multiply $- 1$ by $9$ .

$3 (\frac{125}{3} - 9) + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.5

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 (\frac{125}{3} - 9 \cdot \frac{3}{3}) + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.6

Combine $- 9$ and $\frac{3}{3}$ .

$3 (\frac{125}{3} + \frac{- 9 \cdot 3}{3}) + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.7

Combine the numerators over the common denominator.

$3 \frac{125 - 9 \cdot 3}{3} + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.8

Simplify the numerator.

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Step 9.4.8.1

Multiply $- 9$ by $3$ .

$3 \frac{125 - 27}{3} + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.8.2

Subtract $27$ from $125$ .

$3 (\frac{98}{3}) + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.9

Combine $3$ and $\frac{98}{3}$ .

$\frac{3 \cdot 98}{3} + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.10

Multiply $3$ by $98$ .

$\frac{294}{3} + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.11

Cancel the common factor of $294$ and $3$ .

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Step 9.4.11.1

Factor $3$ out of $294$ .

$\frac{3 \cdot 98}{3} + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.11.2

Cancel the common factors.

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Step 9.4.11.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 98}{3 (1)} + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.11.2.2

Cancel the common factor.

$\frac{3 \cdot 98}{3 \cdot 1} + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.11.2.3

Rewrite the expression.

$\frac{98}{1} + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.11.2.4

Divide $98$ by $1$ .

$98 + 2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.12

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

$98 + 2 (\frac{25}{2} - \frac{3^{2}}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.13

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

$98 + 2 (\frac{25}{2} - \frac{9}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.14

Combine the numerators over the common denominator.

$98 + 2 \frac{25 - 9}{2} + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.15

Subtract $9$ from $25$ .

$98 + 2 (\frac{16}{2}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.16

Cancel the common factor of $16$ and $2$ .

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Step 9.4.16.1

Factor $2$ out of $16$ .

$98 + 2 \frac{2 \cdot 8}{2} + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.16.2

Cancel the common factors.

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Step 9.4.16.2.1

Factor $2$ out of $2$ .

$98 + 2 \frac{2 \cdot 8}{2 (1)} + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.16.2.2

Cancel the common factor.

$98 + 2 \frac{2 \cdot 8}{2 \cdot 1} + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.16.2.3

Rewrite the expression.

$98 + 2 (\frac{8}{1}) + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.16.2.4

Divide $8$ by $1$ .

$98 + 2 \cdot 8 + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.17

Multiply $2$ by $8$ .

$98 + 16 + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.18

Add $98$ and $16$ .

$114 + (- 1 \cdot 5) + 1 \cdot 3$

Step 9.4.19

Multiply $- 1$ by $5$ .

$114 - 5 + 1 \cdot 3$

Step 9.4.20

Multiply $3$ by $1$ .

$114 - 5 + 3$

Step 9.4.21

Add $- 5$ and $3$ .

$114 - 2$

Step 9.4.22

Subtract $2$ from $114$ .

$112$

Step 10

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