Calculus Examples

$\int_{0}^{2} (3 x^{2} - \sin (x)) d x$

Step 1

Remove parentheses.

$\int_{0}^{2} 3 x^{2} - \sin (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{2} 3 x^{2} d x + \int_{0}^{2} - \sin (x) d x$

Step 3

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

$3 \int_{0}^{2} x^{2} d x + \int_{0}^{2} - \sin (x) d x$

Step 4

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

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

Step 5

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

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

Step 6

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

$3 (\frac{x^{3}}{3}]_{0}^{2}) - \int_{0}^{2} \sin (x) d x$

Step 7

The integral of $\sin (x)$ with respect to $x$ is $- \cos (x)$ .

$3 (\frac{x^{3}}{3}]_{0}^{2}) - (- \cos (x)]_{0}^{2})$

Step 8

Simplify the answer.

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Step 8.1

Substitute and simplify.

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Step 8.1.1

Evaluate $\frac{x^{3}}{3}$ at $2$ and at $0$ .

$3 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3}) - (- \cos (x)]_{0}^{2})$

Step 8.1.2

Evaluate $- \cos (x)$ at $2$ and at $0$ .

$3 (\frac{2^{3}}{3} - \frac{0^{3}}{3}) - (- \cos (2) + \cos (0))$

Step 8.1.3

Simplify.

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Step 8.1.3.1

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

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

Step 8.1.3.2

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

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

Step 8.1.3.3

Cancel the common factor of $0$ and $3$ .

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Step 8.1.3.3.1

Factor $3$ out of $0$ .

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

Step 8.1.3.3.2

Cancel the common factors.

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Step 8.1.3.3.2.1

Factor $3$ out of $3$ .

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

Step 8.1.3.3.2.2

Cancel the common factor.

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

Step 8.1.3.3.2.3

Rewrite the expression.

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

Step 8.1.3.3.2.4

Divide $0$ by $1$ .

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

Step 8.1.3.4

Multiply $- 1$ by $0$ .

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

Step 8.1.3.5

Add $\frac{8}{3}$ and $0$ .

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

Step 8.1.3.6

Combine $3$ and $\frac{8}{3}$ .

$\frac{3 \cdot 8}{3} - (- \cos (2) + \cos (0))$

Step 8.1.3.7

Multiply $3$ by $8$ .

$\frac{24}{3} - (- \cos (2) + \cos (0))$

Step 8.1.3.8

Cancel the common factor of $24$ and $3$ .

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Step 8.1.3.8.1

Factor $3$ out of $24$ .

$\frac{3 \cdot 8}{3} - (- \cos (2) + \cos (0))$

Step 8.1.3.8.2

Cancel the common factors.

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Step 8.1.3.8.2.1

Factor $3$ out of $3$ .

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

Step 8.1.3.8.2.2

Cancel the common factor.

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

Step 8.1.3.8.2.3

Rewrite the expression.

$\frac{8}{1} - (- \cos (2) + \cos (0))$

Step 8.1.3.8.2.4

Divide $8$ by $1$ .

$8 - (- \cos (2) + \cos (0))$

Step 8.2

The exact value of $\cos (0)$ is $1$ .

$8 - (- \cos (2) + 1)$

Step 8.3

Simplify.

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Step 8.3.1

Evaluate $\cos (2)$ .

$8 - (- - 0.41614683 + 1)$

Step 8.3.2

Multiply $- 1$ by $- 0.41614683$ .

$8 - (0.41614683 + 1)$

Step 8.3.3

Add $0.41614683$ and $1$ .

$8 - 1 \cdot 1.41614683$

Step 8.3.4

Multiply $- 1$ by $1.41614683$ .

$8 - 1.41614683$

Step 8.3.5

Subtract $1.41614683$ from $8$ .

$6.58385316$