Calculus Examples

$\int \sqrt{5 + 4 x - x^{2}} d x$

Step 1

Complete the square.

Tap for more steps...

Step 1.1

Simplify the expression.

Tap for more steps...

Step 1.1.1

Move $5$ .

$4 x - x^{2} + 5$

Step 1.1.2

Reorder $4 x$ and $- x^{2}$ .

$- x^{2} + 4 x + 5$

Step 1.2

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 1$

$b = 4$

$c = 5$

Step 1.3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Tap for more steps...

Step 1.4.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{4}{2 \cdot - 1}$

Step 1.4.2

Simplify the right side.

Tap for more steps...

Step 1.4.2.1

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

Tap for more steps...

Step 1.4.2.1.1

Factor $2$ out of $4$ .

$d = \frac{2 \cdot 2}{2 \cdot - 1}$

Step 1.4.2.1.2

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$d = - 1 \cdot 2$

Step 1.4.2.2

Multiply $- 1$ by $2$ .

$d = - 2$

Step 1.5

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Tap for more steps...

Step 1.5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 5 - \frac{4^{2}}{4 \cdot - 1}$

Step 1.5.2

Simplify the right side.

Tap for more steps...

Step 1.5.2.1

Simplify each term.

Tap for more steps...

Step 1.5.2.1.1

Cancel the common factor of $4^{2}$ and $4$ .

Tap for more steps...

Step 1.5.2.1.1.1

Factor $4$ out of $4^{2}$ .

$e = 5 - \frac{4 \cdot 4}{4 \cdot - 1}$

Step 1.5.2.1.1.2

Move the negative one from the denominator of $\frac{4}{- 1}$ .

$e = 5 - (- 1 \cdot 4)$

Step 1.5.2.1.2

Multiply $- (- 1 \cdot 4)$ .

Tap for more steps...

Step 1.5.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 5 - - 4$

Step 1.5.2.1.2.2

Multiply $- 1$ by $- 4$ .

$e = 5 + 4$

Step 1.5.2.2

Add $5$ and $4$ .

$e = 9$

Step 1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - 2)}^{2} + 9$ .

$\int \sqrt{- {(x - 2)}^{2} + 9} d x$

Step 2

Let $u_{1} = x - 2$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 2.1

Let $u_{1} = x - 2$ . Find $\frac{d u_{1}}{d x}$ .

Tap for more steps...

Step 2.1.1

Differentiate $x - 2$ .

$\frac{d}{d x} [x - 2]$

Step 2.1.2

By the Sum Rule, the derivative of $x - 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 2.1.3

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

$1 + \frac{d}{d x} [- 2]$

Step 2.1.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.1.5

Add $1$ and $0$ .

$1$

Step 2.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \sqrt{- {u_{1}}^{2} + 9} d u_{1}$

Step 3

Let $u_{1} = 3 \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d u_{1} = 3 \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $3 \cos (t)$ is positive.

$\int \sqrt{- {(3 \sin (t))}^{2} + 9} (3 \cos (t)) d t$

Step 4

Simplify terms.

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

Simplify each term.

Tap for more steps...

Step 4.1.1.1

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

$\int \sqrt{- (3^{2} \sin^{2} (t)) + 9} (3 \cos (t)) d t$

Step 4.1.1.2

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

$\int \sqrt{- (9 \sin^{2} (t)) + 9} (3 \cos (t)) d t$

Step 4.1.1.3

Multiply $9$ by $- 1$ .

$\int \sqrt{- 9 \sin^{2} (t) + 9} (3 \cos (t)) d t$

Step 4.1.2

Reorder $- 9 \sin^{2} (t)$ and $9$ .

$\int \sqrt{9 - 9 \sin^{2} (t)} (3 \cos (t)) d t$

Step 4.1.3

Factor $9$ out of $9$ .

$\int \sqrt{9 (1) - 9 \sin^{2} (t)} (3 \cos (t)) d t$

Step 4.1.4

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

$\int \sqrt{9 (1) + 9 (- \sin^{2} (t))} (3 \cos (t)) d t$

Step 4.1.5

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

$\int \sqrt{9 (1 - \sin^{2} (t))} (3 \cos (t)) d t$

Step 4.1.6

Apply pythagorean identity.

$\int \sqrt{9 \cos^{2} (t)} (3 \cos (t)) d t$

Step 4.1.7

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

$\int \sqrt{{(3 \cos (t))}^{2}} (3 \cos (t)) d t$

Step 4.1.8

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

$\int 3 \cos (t) (3 \cos (t)) d t$

Step 4.2

Simplify.

Tap for more steps...

Step 4.2.1

Multiply $3$ by $3$ .

$\int 9 \cos (t) \cos (t) d t$

Step 4.2.2

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

$\int 9 (\cos^{1} (t) \cos (t)) d t$

Step 4.2.3

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

$\int 9 (\cos^{1} (t) \cos^{1} (t)) d t$

Step 4.2.4

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

$\int 9 {\cos (t)}^{1 + 1} d t$

Step 4.2.5

Add $1$ and $1$ .

$\int 9 \cos^{2} (t) d t$

Step 5

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

$9 \int \cos^{2} (t) d t$

Step 6

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

$9 \int \frac{1 + \cos (2 t)}{2} d t$

Step 7

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

$9 (\frac{1}{2} \int 1 + \cos (2 t) d t)$

Step 8

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

$\frac{9}{2} \int 1 + \cos (2 t) d t$

Step 9

Split the single integral into multiple integrals.

$\frac{9}{2} (\int d t + \int \cos (2 t) d t)$

Step 10

Apply the constant rule.

$\frac{9}{2} (t + C + \int \cos (2 t) d t)$

Step 11

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

Tap for more steps...

Step 11.1

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

Tap for more steps...

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

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{9}{2} (t + C + \int \cos (u_{2}) \frac{1}{2} d u_{2})$

Step 12

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

$\frac{9}{2} (t + C + \int \frac{\cos (u_{2})}{2} d u_{2})$

Step 13

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

$\frac{9}{2} (t + C + \frac{1}{2} \int \cos (u_{2}) d u_{2})$

Step 14

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{9}{2} (t + C + \frac{1}{2} (\sin (u_{2}) + C))$

Step 15

Simplify.

$\frac{9}{2} (t + \frac{1}{2} \sin (u_{2})) + C$

Step 16

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 16.1

Replace all occurrences of $t$ with $\arcsin (\frac{u_{1}}{3})$ .

$\frac{9}{2} (\arcsin (\frac{u_{1}}{3}) + \frac{1}{2} \sin (u_{2})) + C$

Step 16.2

Replace all occurrences of $u_{2}$ with $2 t$ .

$\frac{9}{2} (\arcsin (\frac{u_{1}}{3}) + \frac{1}{2} \sin (2 t)) + C$

Step 16.3

Replace all occurrences of $u_{1}$ with $x - 2$ .

$\frac{9}{2} (\arcsin (\frac{x - 2}{3}) + \frac{1}{2} \sin (2 t)) + C$

Step 16.4

Replace all occurrences of $t$ with $\arcsin (\frac{u_{1}}{3})$ .

$\frac{9}{2} (\arcsin (\frac{x - 2}{3}) + \frac{1}{2} \sin (2 \arcsin (\frac{u_{1}}{3}))) + C$

Step 16.5

Replace all occurrences of $u_{1}$ with $x - 2$ .

$\frac{9}{2} (\arcsin (\frac{x - 2}{3}) + \frac{1}{2} \sin (2 \arcsin (\frac{x - 2}{3}))) + C$

Step 17

Simplify.

Tap for more steps...

Step 17.1

Combine $\frac{1}{2}$ and $\sin (2 \arcsin (\frac{x - 2}{3}))$ .

$\frac{9}{2} (\arcsin (\frac{x - 2}{3}) + \frac{\sin (2 \arcsin (\frac{x - 2}{3}))}{2}) + C$

Step 17.2

Apply the distributive property.

$\frac{9}{2} \arcsin (\frac{x - 2}{3}) + \frac{9}{2} \cdot \frac{\sin (2 \arcsin (\frac{x - 2}{3}))}{2} + C$

Step 17.3

Combine $\frac{9}{2}$ and $\arcsin (\frac{x - 2}{3})$ .

$\frac{9 \arcsin (\frac{x - 2}{3})}{2} + \frac{9}{2} \cdot \frac{\sin (2 \arcsin (\frac{x - 2}{3}))}{2} + C$

Step 17.4

Multiply $\frac{9}{2} \cdot \frac{\sin (2 \arcsin (\frac{x - 2}{3}))}{2}$ .

Tap for more steps...

Step 17.4.1

Multiply $\frac{9}{2}$ by $\frac{\sin (2 \arcsin (\frac{x - 2}{3}))}{2}$ .

$\frac{9 \arcsin (\frac{x - 2}{3})}{2} + \frac{9 \sin (2 \arcsin (\frac{x - 2}{3}))}{2 \cdot 2} + C$

Step 17.4.2

Multiply $2$ by $2$ .

$\frac{9 \arcsin (\frac{x - 2}{3})}{2} + \frac{9 \sin (2 \arcsin (\frac{x - 2}{3}))}{4} + C$

Step 18

Reorder terms.

$\frac{9}{2} \arcsin (\frac{1}{3} (x - 2)) + \frac{9}{4} \sin (2 \arcsin (\frac{1}{3} (x - 2))) + C$