Calculus Examples

$\int \frac{x + 5}{\sqrt{9 - {(x - 3)}^{2}}} d x$

Step 1

Let $u = x - 3$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x - 3$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x - 3$ .

$\frac{d}{d x} [x - 3]$

Step 1.1.2

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$

Step 1.1.3

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

$1 + \frac{d}{d x} [- 3]$

Step 1.1.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{u + 3 + 5}{\sqrt{9 - u^{2}}} d u$

Step 2

Add $3$ and $5$ .

$\int \frac{u + 8}{\sqrt{9 - u^{2}}} d u$

Step 3

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

$\int \frac{3 \sin (t) + 8}{\sqrt{9 - {(3 \sin (t))}^{2}}} (3 \cos (t)) d t$

Step 4

Simplify terms.

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Step 4.1

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

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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 $3 \sin (t)$ .

$\int \frac{3 \sin (t) + 8}{\sqrt{9 - (3^{2} \sin^{2} (t))}} (3 \cos (t)) d t$

Step 4.1.1.2

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

$\int \frac{3 \sin (t) + 8}{\sqrt{9 - (9 \sin^{2} (t))}} (3 \cos (t)) d t$

Step 4.1.1.3

Multiply $9$ by $- 1$ .

$\int \frac{3 \sin (t) + 8}{\sqrt{9 - 9 \sin^{2} (t)}} (3 \cos (t)) d t$

Step 4.1.2

Factor $9$ out of $9$ .

$\int \frac{3 \sin (t) + 8}{\sqrt{9 (1) - 9 \sin^{2} (t)}} (3 \cos (t)) d t$

Step 4.1.3

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

$\int \frac{3 \sin (t) + 8}{\sqrt{9 (1) + 9 (- \sin^{2} (t))}} (3 \cos (t)) d t$

Step 4.1.4

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

$\int \frac{3 \sin (t) + 8}{\sqrt{9 (1 - \sin^{2} (t))}} (3 \cos (t)) d t$

Step 4.1.5

Apply pythagorean identity.

$\int \frac{3 \sin (t) + 8}{\sqrt{9 \cos^{2} (t)}} (3 \cos (t)) d t$

Step 4.1.6

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

$\int \frac{3 \sin (t) + 8}{\sqrt{{(3 \cos (t))}^{2}}} (3 \cos (t)) d t$

Step 4.1.7

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

$\int \frac{3 \sin (t) + 8}{3 \cos (t)} (3 \cos (t)) d t$

Step 4.2

Cancel the common factor of $3 \cos (t)$ .

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Step 4.2.1

Cancel the common factor.

$\int \frac{3 \sin (t) + 8}{3 \cos (t)} (3 \cos (t)) d t$

Step 4.2.2

Rewrite the expression.

$\int 3 \sin (t) + 8 d t$

Step 5

Split the single integral into multiple integrals.

$\int 3 \sin (t) d t + \int 8 d t$

Step 6

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

$3 \int \sin (t) d t + \int 8 d t$

Step 7

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

$3 (- \cos (t) + C) + \int 8 d t$

Step 8

Apply the constant rule.

$3 (- \cos (t) + C) + 8 t + C$

Step 9

Simplify.

$- 3 \cos (t) + 8 t + C$

Step 10

Substitute back in for each integration substitution variable.

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Step 10.1

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

$- 3 \cos (\arcsin (\frac{u}{3})) + 8 \arcsin (\frac{u}{3}) + C$

Step 10.2

Replace all occurrences of $u$ with $x - 3$ .

$- 3 \cos (\arcsin (\frac{x - 3}{3})) + 8 \arcsin (\frac{x - 3}{3}) + C$

Step 11

Reorder terms.

$- 3 \cos (\arcsin (\frac{1}{3} (x - 3))) + 8 \arcsin (\frac{1}{3} (x - 3)) + C$