Calculus Examples

$\sin^{-1} (d x)$

Step 1

Write $\sin^{-1} (d x)$ as a function.

$f (d) = \sin^{-1} (d x)$

Step 2

The function $F (d)$ can be found by finding the indefinite integral of the derivative $f (d)$ .

$F (d) = \int f (d) d \cdot d$

Step 3

Set up the integral to solve.

$F (d) = \int \sin^{-1} (d x) d \cdot d$

Step 4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sin^{-1} (d x)$ and $d v = 1$ .

$\sin^{-1} (d x) d - \int d \frac{x}{\sqrt{1 - d^{2} x^{2}}} d \cdot d$

Step 5

Combine $d$ and $\frac{x}{\sqrt{1 - d^{2} x^{2}}}$ .

$\sin^{-1} (d x) d - \int \frac{d x}{\sqrt{1 - d^{2} x^{2}}} d \cdot d$

Step 6

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

$\sin^{-1} (d x) d - (x \int \frac{d}{\sqrt{1 - d^{2} x^{2}}} d \cdot d)$

Step 7

Let $u = 1 - d^{2} x^{2}$ . Then $d u = - 2 \cdot d x^{2} d \cdot d$ , so $\frac{1}{- 2 x^{2}} d u = d \cdot d \cdot d$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 7.1

Let $u = 1 - d^{2} x^{2}$ . Find $\frac{d u}{d \cdot d}$ .

Tap for more steps...

Step 7.1.1

Differentiate $1 - d^{2} x^{2}$ .

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

Step 7.1.2

Differentiate.

Tap for more steps...

Step 7.1.2.1

By the Sum Rule, the derivative of $1 - d^{2} x^{2}$ with respect to $d$ is $\frac{d}{d \cdot d} [1] + \frac{d}{d \cdot d} [- d^{2} x^{2}]$ .

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

Step 7.1.2.2

Since $1$ is constant with respect to $d$ , the derivative of $1$ with respect to $d$ is $0$ .

$0 + \frac{d}{d \cdot d} [- d^{2} x^{2}]$

Step 7.1.3

Evaluate $\frac{d}{d \cdot d} [- d^{2} x^{2}]$ .

Tap for more steps...

Step 7.1.3.1

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

$0 - x^{2} \frac{d}{d \cdot d} [d^{2}]$

Step 7.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d \cdot d} [d^{n}]$ is $n d^{n - 1}$ where $n = 2$ .

$0 - x^{2} (2 d)$

Step 7.1.3.3

Multiply $2$ by $- 1$ .

$0 - 2 x^{2} d$

Step 7.1.4

Subtract $2 x^{2} d$ from $0$ .

$- 2 x^{2} d$

Step 7.1.5

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

$d (- 2 x^{2})$

Step 7.1.6

Reorder $d$ and $- 2$ .

$- 2 \cdot d x^{2}$

Step 7.2

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

$\sin^{-1} (d x) d - x \int \frac{1}{\sqrt{u}} \cdot \frac{1}{- 2 x^{2}} d u$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Move the negative in front of the fraction.

$\sin^{-1} (d x) d - x \int \frac{1}{\sqrt{u}} (- \frac{1}{2 x^{2}}) d u$

Step 8.2

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{2 x^{2}}$ .

$\sin^{-1} (d x) d - x \int - \frac{1}{\sqrt{u} (2 x^{2})} d u$

Step 8.3

Move $2$ to the left of $\sqrt{u}$ .

$\sin^{-1} (d x) d - x \int - \frac{1}{2 \sqrt{u} x^{2}} d u$

Step 9

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

$\sin^{-1} (d x) d - x (- \int \frac{1}{2 \sqrt{u} x^{2}} d u)$

Step 10

Simplify.

Tap for more steps...

Step 10.1

Multiply $- 1$ by $- 1$ .

$\sin^{-1} (d x) d + 1 x \int \frac{1}{2 \sqrt{u} x^{2}} d u$

Step 10.2

Multiply $x$ by $1$ .

$\sin^{-1} (d x) d + x \int \frac{1}{2 \sqrt{u} x^{2}} d u$

Step 11

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

$\sin^{-1} (d x) d + x (\frac{1}{2 x^{2}} \int \frac{1}{\sqrt{u}} d u)$

Step 12

Simplify the expression.

Tap for more steps...

Step 12.1

Simplify.

Tap for more steps...

Step 12.1.1

Combine $\frac{1}{2 x^{2}}$ and $x$ .

$\sin^{-1} (d x) d + \frac{x}{2 x^{2}} \int \frac{1}{\sqrt{u}} d u$

Step 12.1.2

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

Tap for more steps...

Step 12.1.2.1

Raise $x$ to the power of $1$ .

$\sin^{-1} (d x) d + \frac{x^{1}}{2 x^{2}} \int \frac{1}{\sqrt{u}} d u$

Step 12.1.2.2

Factor $x$ out of $x^{1}$ .

$\sin^{-1} (d x) d + \frac{x \cdot 1}{2 x^{2}} \int \frac{1}{\sqrt{u}} d u$

Step 12.1.2.3

Cancel the common factors.

Tap for more steps...

Step 12.1.2.3.1

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

$\sin^{-1} (d x) d + \frac{x \cdot 1}{x (2 x)} \int \frac{1}{\sqrt{u}} d u$

Step 12.1.2.3.2

Cancel the common factor.

$\sin^{-1} (d x) d + \frac{x \cdot 1}{x (2 x)} \int \frac{1}{\sqrt{u}} d u$

Step 12.1.2.3.3

Rewrite the expression.

$\sin^{-1} (d x) d + \frac{1}{2 x} \int \frac{1}{\sqrt{u}} d u$

Step 12.2

Apply basic rules of exponents.

Tap for more steps...

Step 12.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\sin^{-1} (d x) d + \frac{1}{2 x} \int \frac{1}{u^{\frac{1}{2}}} d u$

Step 12.2.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\sin^{-1} (d x) d + \frac{1}{2 x} \int {(u^{\frac{1}{2}})}^{- 1} d u$

Step 12.2.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

Tap for more steps...

Step 12.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin^{-1} (d x) d + \frac{1}{2 x} \int u^{\frac{1}{2} \cdot - 1} d u$

Step 12.2.3.2

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

$\sin^{-1} (d x) d + \frac{1}{2 x} \int u^{\frac{- 1}{2}} d u$

Step 12.2.3.3

Move the negative in front of the fraction.

$\sin^{-1} (d x) d + \frac{1}{2 x} \int u^{- \frac{1}{2}} d u$

Step 13

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

$\sin^{-1} (d x) d + \frac{1}{2 x} (2 u^{\frac{1}{2}} + C)$

Step 14

Simplify.

Tap for more steps...

Step 14.1

Rewrite $\sin^{-1} (d x) d + \frac{1}{2 x} (2 u^{\frac{1}{2}} + C)$ as $\sin^{-1} (d x) d + 2 \frac{1}{2 x} u^{\frac{1}{2}} + C$ .

$\sin^{-1} (d x) d + 2 \frac{1}{2 x} u^{\frac{1}{2}} + C$

Step 14.2

Simplify.

Tap for more steps...

Step 14.2.1

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

$\sin^{-1} (d x) d + \frac{2}{2 x} u^{\frac{1}{2}} + C$

Step 14.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 14.2.2.1

Cancel the common factor.

$\sin^{-1} (d x) d + \frac{2}{2 x} u^{\frac{1}{2}} + C$

Step 14.2.2.2

Rewrite the expression.

$\sin^{-1} (d x) d + \frac{1}{x} u^{\frac{1}{2}} + C$

Step 14.2.3

Combine $\frac{1}{x}$ and $u^{\frac{1}{2}}$ .

$\sin^{-1} (d x) d + \frac{u^{\frac{1}{2}}}{x} + C$

Step 15

Replace all occurrences of $u$ with $1 - d^{2} x^{2}$ .

$\sin^{-1} (d x) d + \frac{{(1 - d^{2} x^{2})}^{\frac{1}{2}}}{x} + C$

Step 16

The answer is the antiderivative of the function $f (d) = \sin^{-1} (d x)$ .

$F (d) =$ $\sin^{-1} (d x) d + \frac{{(1 - d^{2} x^{2})}^{\frac{1}{2}}}{x} + C$