Calculus Examples

$\int_{- 3}^{0} (1 + \sqrt{9 - x^{2}}) d x$

Step 1

Remove parentheses.

$\int_{- 3}^{0} 1 + \sqrt{9 - x^{2}} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 3}^{0} d x + \int_{- 3}^{0} \sqrt{9 - x^{2}} d x$

Step 3

Apply the constant rule.

$x]_{- 3}^{0} + \int_{- 3}^{0} \sqrt{9 - x^{2}} d x$

Step 4

Let $x = 3 \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = 3 \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $3 \cos (t)$ is positive.

$x]_{- 3}^{0} + \int_{- \frac{π}{2}}^{0} \sqrt{9 - {(3 \sin (t))}^{2}} (3 \cos (t)) d t$

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Simplify $\sqrt{9 - {(3 \sin (t))}^{2}}$ .

Tap for more steps...

Step 5.1.1

Simplify each term.

Tap for more steps...

Step 5.1.1.1

Apply the product rule to $3 \sin (t)$ .

$x]_{- 3}^{0} + \int_{- \frac{π}{2}}^{0} \sqrt{9 - (3^{2} \sin^{2} (t))} (3 \cos (t)) d t$

Step 5.1.1.2

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

$x]_{- 3}^{0} + \int_{- \frac{π}{2}}^{0} \sqrt{9 - (9 \sin^{2} (t))} (3 \cos (t)) d t$

Step 5.1.1.3

Multiply $9$ by $- 1$ .

$x]_{- 3}^{0} + \int_{- \frac{π}{2}}^{0} \sqrt{9 - 9 \sin^{2} (t)} (3 \cos (t)) d t$

Step 5.1.2

Factor $9$ out of $9$ .

$x]_{- 3}^{0} + \int_{- \frac{π}{2}}^{0} \sqrt{9 (1) - 9 \sin^{2} (t)} (3 \cos (t)) d t$

Step 5.1.3

Factor $9$ out of $- 9 \sin^{2} (t)$ .

$x]_{- 3}^{0} + \int_{- \frac{π}{2}}^{0} \sqrt{9 (1) + 9 (- \sin^{2} (t))} (3 \cos (t)) d t$

Step 5.1.4

Factor $9$ out of $9 (1) + 9 (- \sin^{2} (t))$ .

$x]_{- 3}^{0} + \int_{- \frac{π}{2}}^{0} \sqrt{9 (1 - \sin^{2} (t))} (3 \cos (t)) d t$

Step 5.1.5

Apply pythagorean identity.

$x]_{- 3}^{0} + \int_{- \frac{π}{2}}^{0} \sqrt{9 \cos^{2} (t)} (3 \cos (t)) d t$

Step 5.1.6

Rewrite $9 \cos^{2} (t)$ as ${(3 \cos (t))}^{2}$ .

$x]_{- 3}^{0} + \int_{- \frac{π}{2}}^{0} \sqrt{{(3 \cos (t))}^{2}} (3 \cos (t)) d t$

Step 5.1.7

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

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

Step 5.2

Simplify.

Tap for more steps...

Step 5.2.1

Multiply $3$ by $3$ .

$x]_{- 3}^{0} + \int_{- \frac{π}{2}}^{0} 9 \cos (t) \cos (t) d t$

Step 5.2.2

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

$x]_{- 3}^{0} + \int_{- \frac{π}{2}}^{0} 9 (\cos^{1} (t) \cos (t)) d t$

Step 5.2.3

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

$x]_{- 3}^{0} + \int_{- \frac{π}{2}}^{0} 9 (\cos^{1} (t) \cos^{1} (t)) d t$

Step 5.2.4

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

$x]_{- 3}^{0} + \int_{- \frac{π}{2}}^{0} 9 {\cos (t)}^{1 + 1} d t$

Step 5.2.5

Add $1$ and $1$ .

$x]_{- 3}^{0} + \int_{- \frac{π}{2}}^{0} 9 \cos^{2} (t) d t$

Step 6

Since $9$ is constant with respect to $t$ , move $9$ out of the integral.

$x]_{- 3}^{0} + 9 \int_{- \frac{π}{2}}^{0} \cos^{2} (t) d t$

Step 7

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

$x]_{- 3}^{0} + 9 \int_{- \frac{π}{2}}^{0} \frac{1 + \cos (2 t)}{2} d t$

Step 8

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

$x]_{- 3}^{0} + 9 (\frac{1}{2} \int_{- \frac{π}{2}}^{0} 1 + \cos (2 t) d t)$

Step 9

Combine $\frac{1}{2}$ and $9$ .

$x]_{- 3}^{0} + \frac{9}{2} \int_{- \frac{π}{2}}^{0} 1 + \cos (2 t) d t$

Step 10

Split the single integral into multiple integrals.

$x]_{- 3}^{0} + \frac{9}{2} (\int_{- \frac{π}{2}}^{0} d t + \int_{- \frac{π}{2}}^{0} \cos (2 t) d t)$

Step 11

Apply the constant rule.

$x]_{- 3}^{0} + \frac{9}{2} (t]_{- \frac{π}{2}}^{0} + \int_{- \frac{π}{2}}^{0} \cos (2 t) d t)$

Step 12

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

Tap for more steps...

Step 12.1

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

Tap for more steps...

Step 12.1.1

Differentiate $2 t$ .

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

Step 12.1.2

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 12.1.3

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 12.1.4

Multiply $2$ by $1$ .

$2$

Step 12.2

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

$u_{lower} = 2 (- \frac{π}{2})$

Step 12.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 12.3.1

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

$u_{lower} = 2 \frac{- π}{2}$

Step 12.3.2

Cancel the common factor.

$u_{lower} = 2 \frac{- π}{2}$

Step 12.3.3

Rewrite the expression.

$u_{lower} = - π$

Step 12.4

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

$u_{upper} = 2 \cdot 0$

Step 12.5

Multiply $2$ by $0$ .

$u_{upper} = 0$

Step 12.6

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

$u_{lower} = - π$

$u_{upper} = 0$

Step 12.7

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

$x]_{- 3}^{0} + \frac{9}{2} (t]_{- \frac{π}{2}}^{0} + \int_{- π}^{0} \cos (u) \frac{1}{2} d u)$

Step 13

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

$x]_{- 3}^{0} + \frac{9}{2} (t]_{- \frac{π}{2}}^{0} + \int_{- π}^{0} \frac{\cos (u)}{2} d u)$

Step 14

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

$x]_{- 3}^{0} + \frac{9}{2} (t]_{- \frac{π}{2}}^{0} + \frac{1}{2} \int_{- π}^{0} \cos (u) d u)$

Step 15

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

$x]_{- 3}^{0} + \frac{9}{2} (t]_{- \frac{π}{2}}^{0} + \frac{1}{2} \sin (u)]_{- π}^{0})$

Step 16

Substitute and simplify.

Tap for more steps...

Step 16.1

Evaluate $x$ at $0$ and at $- 3$ .

$(0) + 3 + \frac{9}{2} (t]_{- \frac{π}{2}}^{0} + \frac{1}{2} \sin (u)]_{- π}^{0})$

Step 16.2

Evaluate $t$ at $0$ and at $- \frac{π}{2}$ .

$(0) + 3 + \frac{9}{2} ((0) + \frac{π}{2} + \frac{1}{2} \sin (u)]_{- π}^{0})$

Step 16.3

Evaluate $\sin (u)$ at $0$ and at $- π$ .

$(0) + 3 + \frac{9}{2} ((0) + \frac{π}{2} + \frac{1}{2} ((\sin (0)) - \sin (- π)))$

Step 16.4

Simplify.

Tap for more steps...

Step 16.4.1

Add $0$ and $3$ .

$3 + \frac{9}{2} ((0) + \frac{π}{2} + \frac{1}{2} ((\sin (0)) - \sin (- π)))$

Step 16.4.2

Add $0$ and $\frac{π}{2}$ .

$3 + \frac{9}{2} (\frac{π}{2} + \frac{1}{2} (\sin (0) - \sin (- π)))$

Step 17

Simplify.

Tap for more steps...

Step 17.1

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

$3 + \frac{9}{2} (\frac{π}{2} + \frac{1}{2} (0 - \sin (- π)))$

Step 17.2

Subtract $\sin (- π)$ from $0$ .

$3 + \frac{9}{2} (\frac{π}{2} + \frac{1}{2} (- \sin (- π)))$

Step 17.3

Combine $\frac{1}{2}$ and $\sin (- π)$ .

$3 + \frac{9}{2} (\frac{π}{2} - \frac{\sin (- π)}{2})$

Step 18

Simplify.

Tap for more steps...

Step 18.1

Simplify the numerator.

Tap for more steps...

Step 18.1.1

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

$3 + \frac{9}{2} (\frac{π}{2} - \frac{\sin (0)}{2})$

Step 18.1.2

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

$3 + \frac{9}{2} (\frac{π}{2} - \frac{0}{2})$

Step 18.2

Divide $0$ by $2$ .

$3 + \frac{9}{2} (\frac{π}{2} - 0)$

Step 18.3

Multiply $- 1$ by $0$ .

$3 + \frac{9}{2} (\frac{π}{2} + 0)$

Step 18.4

Add $\frac{π}{2}$ and $0$ .

$3 + \frac{9}{2} \cdot \frac{π}{2}$

Step 18.5

Multiply $\frac{9}{2} \cdot \frac{π}{2}$ .

Tap for more steps...

Step 18.5.1

Multiply $\frac{9}{2}$ by $\frac{π}{2}$ .

$3 + \frac{9 π}{2 \cdot 2}$

Step 18.5.2

Multiply $2$ by $2$ .

$3 + \frac{9 π}{4}$

Step 19

The result can be shown in multiple forms.

Exact Form:

$3 + \frac{9 π}{4}$

Decimal Form:

$10.06858347 \dots$

Step 20