Calculus Examples

$\int_{0}^{2} (2 x^{3} - 6 x + \frac{3}{x^{2} + 1}) d x$

Step 1

Remove parentheses.

$\int_{0}^{2} 2 x^{3} - 6 x + \frac{3}{x^{2} + 1} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{2} 2 x^{3} d x + \int_{0}^{2} - 6 x d x + \int_{0}^{2} \frac{3}{x^{2} + 1} d x$

Step 3

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

$2 \int_{0}^{2} x^{3} d x + \int_{0}^{2} - 6 x d x + \int_{0}^{2} \frac{3}{x^{2} + 1} d x$

Step 4

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$2 (\frac{1}{4} x^{4}]_{0}^{2}) + \int_{0}^{2} - 6 x d x + \int_{0}^{2} \frac{3}{x^{2} + 1} d x$

Step 5

Combine $\frac{1}{4}$ and $x^{4}$ .

$2 (\frac{x^{4}}{4}]_{0}^{2}) + \int_{0}^{2} - 6 x d x + \int_{0}^{2} \frac{3}{x^{2} + 1} d x$

Step 6

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$2 (\frac{x^{4}}{4}]_{0}^{2}) - 6 \int_{0}^{2} x d x + \int_{0}^{2} \frac{3}{x^{2} + 1} d x$

Step 7

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

$2 (\frac{x^{4}}{4}]_{0}^{2}) - 6 (\frac{1}{2} x^{2}]_{0}^{2}) + \int_{0}^{2} \frac{3}{x^{2} + 1} d x$

Step 8

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

$2 (\frac{x^{4}}{4}]_{0}^{2}) - 6 (\frac{x^{2}}{2}]_{0}^{2}) + \int_{0}^{2} \frac{3}{x^{2} + 1} d x$

Step 9

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

$2 (\frac{x^{4}}{4}]_{0}^{2}) - 6 (\frac{x^{2}}{2}]_{0}^{2}) + 3 \int_{0}^{2} \frac{1}{x^{2} + 1} d x$

Step 10

Simplify the expression.

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Step 10.1

Reorder $x^{2}$ and $1$ .

$2 (\frac{x^{4}}{4}]_{0}^{2}) - 6 (\frac{x^{2}}{2}]_{0}^{2}) + 3 \int_{0}^{2} \frac{1}{1 + x^{2}} d x$

Step 10.2

Rewrite $1$ as $1^{2}$ .

$2 (\frac{x^{4}}{4}]_{0}^{2}) - 6 (\frac{x^{2}}{2}]_{0}^{2}) + 3 \int_{0}^{2} \frac{1}{1^{2} + x^{2}} d x$

Step 11

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x)]_{0}^{2}$ .

$2 (\frac{x^{4}}{4}]_{0}^{2}) - 6 (\frac{x^{2}}{2}]_{0}^{2}) + 3 (\arctan (x)]_{0}^{2})$

Step 12

Substitute and simplify.

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Step 12.1

Evaluate $\frac{x^{4}}{4}$ at $2$ and at $0$ .

$2 ((\frac{2^{4}}{4}) - \frac{0^{4}}{4}) - 6 (\frac{x^{2}}{2}]_{0}^{2}) + 3 (\arctan (x)]_{0}^{2})$

Step 12.2

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

$2 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 (\arctan (x)]_{0}^{2})$

Step 12.3

Evaluate $\arctan (x)$ at $2$ and at $0$ .

$2 (\frac{2^{4}}{4} - \frac{0^{4}}{4}) - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4

Simplify.

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Step 12.4.1

Raise $2$ to the power of $4$ .

$2 (\frac{16}{4} - \frac{0^{4}}{4}) - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.2

Cancel the common factor of $16$ and $4$ .

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Step 12.4.2.1

Factor $4$ out of $16$ .

$2 (\frac{4 \cdot 4}{4} - \frac{0^{4}}{4}) - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.2.2

Cancel the common factors.

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Step 12.4.2.2.1

Factor $4$ out of $4$ .

$2 (\frac{4 \cdot 4}{4 (1)} - \frac{0^{4}}{4}) - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.2.2.2

Cancel the common factor.

$2 (\frac{4 \cdot 4}{4 \cdot 1} - \frac{0^{4}}{4}) - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.2.2.3

Rewrite the expression.

$2 (\frac{4}{1} - \frac{0^{4}}{4}) - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.2.2.4

Divide $4$ by $1$ .

$2 (4 - \frac{0^{4}}{4}) - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.3

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

$2 (4 - \frac{0}{4}) - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.4

Cancel the common factor of $0$ and $4$ .

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Step 12.4.4.1

Factor $4$ out of $0$ .

$2 (4 - \frac{4 (0)}{4}) - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.4.2

Cancel the common factors.

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Step 12.4.4.2.1

Factor $4$ out of $4$ .

$2 (4 - \frac{4 \cdot 0}{4 \cdot 1}) - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.4.2.2

Cancel the common factor.

$2 (4 - \frac{4 \cdot 0}{4 \cdot 1}) - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.4.2.3

Rewrite the expression.

$2 (4 - \frac{0}{1}) - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.4.2.4

Divide $0$ by $1$ .

$2 (4 - 0) - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.5

Multiply $- 1$ by $0$ .

$2 (4 + 0) - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.6

Add $4$ and $0$ .

$2 \cdot 4 - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.7

Multiply $2$ by $4$ .

$8 - 6 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.8

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

$8 - 6 (\frac{4}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.9

Cancel the common factor of $4$ and $2$ .

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Step 12.4.9.1

Factor $2$ out of $4$ .

$8 - 6 (\frac{2 \cdot 2}{2} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.9.2

Cancel the common factors.

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Step 12.4.9.2.1

Factor $2$ out of $2$ .

$8 - 6 (\frac{2 \cdot 2}{2 (1)} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.9.2.2

Cancel the common factor.

$8 - 6 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.9.2.3

Rewrite the expression.

$8 - 6 (\frac{2}{1} - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.9.2.4

Divide $2$ by $1$ .

$8 - 6 (2 - \frac{0^{2}}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.10

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

$8 - 6 (2 - \frac{0}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.11

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

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Step 12.4.11.1

Factor $2$ out of $0$ .

$8 - 6 (2 - \frac{2 (0)}{2}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.11.2

Cancel the common factors.

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Step 12.4.11.2.1

Factor $2$ out of $2$ .

$8 - 6 (2 - \frac{2 \cdot 0}{2 \cdot 1}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.11.2.2

Cancel the common factor.

$8 - 6 (2 - \frac{2 \cdot 0}{2 \cdot 1}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.11.2.3

Rewrite the expression.

$8 - 6 (2 - \frac{0}{1}) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.11.2.4

Divide $0$ by $1$ .

$8 - 6 (2 - 0) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.12

Multiply $- 1$ by $0$ .

$8 - 6 (2 + 0) + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.13

Add $2$ and $0$ .

$8 - 6 \cdot 2 + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.14

Multiply $- 6$ by $2$ .

$8 - 12 + 3 ((\arctan (2)) - \arctan (0))$

Step 12.4.15

Subtract $12$ from $8$ .

$- 4 + 3 (\arctan (2) - \arctan (0))$

Step 13

Simplify.

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Step 13.1

Simplify each term.

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Step 13.1.1

Simplify each term.

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Step 13.1.1.1

Evaluate $\arctan (2)$ .

$- 4 + 3 (1.10714871 - \arctan (0))$

Step 13.1.1.2

The exact value of $\arctan (0)$ is $0$ .

$- 4 + 3 (1.10714871 - 0)$

Step 13.1.1.3

Multiply $- 1$ by $0$ .

$- 4 + 3 (1.10714871 + 0)$

Step 13.1.2

Add $1.10714871$ and $0$ .

$- 4 + 3 \cdot 1.10714871$

Step 13.1.3

Multiply $3$ by $1.10714871$ .

$- 4 + 3.32144615$

Step 13.2

Add $- 4$ and $3.32144615$ .

$- 0.67855384$

Step 14