Calculus Examples

$\int_{- 2}^{2} 3 + \sqrt{4 - x^{2}} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 2}^{2} 3 d x + \int_{- 2}^{2} \sqrt{4 - x^{2}} d x$

Step 2

Apply the constant rule.

$3 x]_{- 2}^{2} + \int_{- 2}^{2} \sqrt{4 - x^{2}} d x$

Step 3

Let $x = 2 \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = 2 \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $2 \cos (t)$ is positive.

$3 x]_{- 2}^{2} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{4 - {(2 \sin (t))}^{2}} (2 \cos (t)) d t$

Step 4

Simplify terms.

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Step 4.1

Simplify $\sqrt{4 - {(2 \sin (t))}^{2}}$ .

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Apply the product rule to $2 \sin (t)$ .

$3 x]_{- 2}^{2} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{4 - (2^{2} \sin^{2} (t))} (2 \cos (t)) d t$

Step 4.1.1.2

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

$3 x]_{- 2}^{2} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{4 - (4 \sin^{2} (t))} (2 \cos (t)) d t$

Step 4.1.1.3

Multiply $4$ by $- 1$ .

$3 x]_{- 2}^{2} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{4 - 4 \sin^{2} (t)} (2 \cos (t)) d t$

Step 4.1.2

Factor $4$ out of $4$ .

$3 x]_{- 2}^{2} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{4 (1) - 4 \sin^{2} (t)} (2 \cos (t)) d t$

Step 4.1.3

Factor $4$ out of $- 4 \sin^{2} (t)$ .

$3 x]_{- 2}^{2} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{4 (1) + 4 (- \sin^{2} (t))} (2 \cos (t)) d t$

Step 4.1.4

Factor $4$ out of $4 (1) + 4 (- \sin^{2} (t))$ .

$3 x]_{- 2}^{2} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{4 (1 - \sin^{2} (t))} (2 \cos (t)) d t$

Step 4.1.5

Apply pythagorean identity.

$3 x]_{- 2}^{2} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{4 \cos^{2} (t)} (2 \cos (t)) d t$

Step 4.1.6

Rewrite $4 \cos^{2} (t)$ as ${(2 \cos (t))}^{2}$ .

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

Step 4.1.7

Pull terms out from under the radical, assuming positive real numbers.

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

Step 4.2

Simplify.

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Step 4.2.1

Multiply $2$ by $2$ .

$3 x]_{- 2}^{2} + \int_{- \frac{π}{2}}^{\frac{π}{2}} 4 \cos (t) \cos (t) d t$

Step 4.2.2

Raise $\cos (t)$ to the power of $1$ .

$3 x]_{- 2}^{2} + \int_{- \frac{π}{2}}^{\frac{π}{2}} 4 (\cos^{1} (t) \cos (t)) d t$

Step 4.2.3

Raise $\cos (t)$ to the power of $1$ .

$3 x]_{- 2}^{2} + \int_{- \frac{π}{2}}^{\frac{π}{2}} 4 (\cos^{1} (t) \cos^{1} (t)) d t$

Step 4.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x]_{- 2}^{2} + \int_{- \frac{π}{2}}^{\frac{π}{2}} 4 {\cos (t)}^{1 + 1} d t$

Step 4.2.5

Add $1$ and $1$ .

$3 x]_{- 2}^{2} + \int_{- \frac{π}{2}}^{\frac{π}{2}} 4 \cos^{2} (t) d t$

Step 5

Since $4$ is constant with respect to $t$ , move $4$ out of the integral.

$3 x]_{- 2}^{2} + 4 \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos^{2} (t) d t$

Step 6

Use the half-angle formula to rewrite $\cos^{2} (t)$ as $\frac{1 + \cos (2 t)}{2}$ .

$3 x]_{- 2}^{2} + 4 \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1 + \cos (2 t)}{2} d t$

Step 7

Since $\frac{1}{2}$ is constant with respect to $t$ , move $\frac{1}{2}$ out of the integral.

$3 x]_{- 2}^{2} + 4 (\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + \cos (2 t) d t)$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{2}$ and $4$ .

$3 x]_{- 2}^{2} + \frac{4}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 8.2

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

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Step 8.2.1

Factor $2$ out of $4$ .

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

Step 8.2.2

Cancel the common factors.

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Step 8.2.2.1

Factor $2$ out of $2$ .

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

Step 8.2.2.2

Cancel the common factor.

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

Step 8.2.2.3

Rewrite the expression.

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

Step 8.2.2.4

Divide $2$ by $1$ .

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

Step 9

Split the single integral into multiple integrals.

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

Step 10

Apply the constant rule.

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

Step 11

Let $u = 2 t$ . Then $d u = 2 d t$ , so $\frac{1}{2} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 11.1

Let $u = 2 t$ . Find $\frac{d u}{d t}$ .

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Step 11.1.1

Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Step 11.1.2

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$2 \frac{d}{d t} [t]$

Step 11.1.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 11.1.4

Multiply $2$ by $1$ .

$2$

Step 11.2

Substitute the lower limit in for $t$ in $u = 2 t$ .

$u_{lower} = 2 (- \frac{π}{2})$

Step 11.3

Cancel the common factor of $2$ .

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Step 11.3.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$u_{lower} = 2 \frac{- π}{2}$

Step 11.3.2

Cancel the common factor.

$u_{lower} = 2 \frac{- π}{2}$

Step 11.3.3

Rewrite the expression.

$u_{lower} = - π$

Step 11.4

Substitute the upper limit in for $t$ in $u = 2 t$ .

$u_{upper} = 2 \frac{π}{2}$

Step 11.5

Cancel the common factor of $2$ .

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Step 11.5.1

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2}$

Step 11.5.2

Rewrite the expression.

$u_{upper} = π$

Step 11.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = - π$

$u_{upper} = π$

Step 11.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

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

Step 12

Combine $\cos (u)$ and $\frac{1}{2}$ .

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

Step 13

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 14

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

$3 x]_{- 2}^{2} + 2 (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{1}{2} \sin (u)]_{- π}^{π})$

Step 15

Combine $\frac{1}{2}$ and $\sin (u)]_{- π}^{π}$ .

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

Step 16

Substitute and simplify.

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Step 16.1

Evaluate $3 x$ at $2$ and at $- 2$ .

$(3 \cdot 2) - 3 \cdot - 2 + 2 (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{\sin (u)]_{- π}^{π}}{2})$

Step 16.2

Evaluate $t$ at $\frac{π}{2}$ and at $- \frac{π}{2}$ .

$(3 \cdot 2) - 3 \cdot - 2 + 2 ((\frac{π}{2}) + \frac{π}{2} + \frac{\sin (u)]_{- π}^{π}}{2})$

Step 16.3

Evaluate $\sin (u)$ at $π$ and at $- π$ .

$(3 \cdot 2) - 3 \cdot - 2 + 2 ((\frac{π}{2}) + \frac{π}{2} + \frac{(\sin (π)) - \sin (- π)}{2})$

Step 16.4

Simplify.

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Step 16.4.1

Multiply $3$ by $2$ .

$6 - 3 \cdot - 2 + 2 ((\frac{π}{2}) + \frac{π}{2} + \frac{(\sin (π)) - \sin (- π)}{2})$

Step 16.4.2

Multiply $- 3$ by $- 2$ .

$6 + 6 + 2 ((\frac{π}{2}) + \frac{π}{2} + \frac{(\sin (π)) - \sin (- π)}{2})$

Step 16.4.3

Add $6$ and $6$ .

$12 + 2 ((\frac{π}{2}) + \frac{π}{2} + \frac{(\sin (π)) - \sin (- π)}{2})$

Step 16.4.4

Combine the numerators over the common denominator.

$12 + 2 (\frac{π + π}{2} + \frac{(\sin (π)) - \sin (- π)}{2})$

Step 16.4.5

Add $π$ and $π$ .

$12 + 2 (\frac{2 π}{2} + \frac{(\sin (π)) - \sin (- π)}{2})$

Step 16.4.6

Cancel the common factor of $2$ .

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Step 16.4.6.1

Cancel the common factor.

$12 + 2 (\frac{2 π}{2} + \frac{(\sin (π)) - \sin (- π)}{2})$

Step 16.4.6.2

Divide $π$ by $1$ .

$12 + 2 (π + \frac{(\sin (π)) - \sin (- π)}{2})$

$12 + 2 (π + \frac{\sin (π) - \sin (- π)}{2})$

Step 17

Simplify.

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Step 17.1

Simplify the numerator.

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Step 17.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$12 + 2 (π + \frac{\sin (0) - \sin (- π)}{2})$

Step 17.1.2

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

$12 + 2 (π + \frac{0 - \sin (- π)}{2})$

Step 17.1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$12 + 2 (π + \frac{0 - \sin (0)}{2})$

Step 17.1.4

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

$12 + 2 (π + \frac{0 - 0}{2})$

Step 17.1.5

Multiply $- 1$ by $0$ .

$12 + 2 (π + \frac{0 + 0}{2})$

Step 17.1.6

Add $0$ and $0$ .

$12 + 2 (π + \frac{0}{2})$

Step 17.2

Divide $0$ by $2$ .

$12 + 2 (π + 0)$

Step 18

Add $π$ and $0$ .

$12 + 2 π$

Step 19

The result can be shown in multiple forms.

Exact Form:

$12 + 2 π$

Decimal Form:

$18.28318530 \dots$

Step 20