Calculus Examples

$\frac{d y}{d x} = \frac{{(2 - y)}^{2}}{2 \sqrt{1 + x}}$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Multiply both sides by $\frac{1}{{(2 - y)}^{2}}$ .

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{1}{{(2 - y)}^{2}} \cdot \frac{{(2 - y)}^{2}}{2 \sqrt{1 + x}}$

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

Combine.

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{1 {(2 - y)}^{2}}{{(2 - y)}^{2} (2 \sqrt{1 + x})}$

Step 1.2.2

Cancel the common factor of ${(2 - y)}^{2}$ .

Tap for more steps...

Step 1.2.2.1

Cancel the common factor.

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{1 {(2 - y)}^{2}}{{(2 - y)}^{2} (2 \sqrt{1 + x})}$

Step 1.2.2.2

Rewrite the expression.

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{1}{2 \sqrt{1 + x}}$

Step 1.2.3

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

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{1}{2 \sqrt{1 + x}} \cdot \frac{\sqrt{1 + x}}{\sqrt{1 + x}}$

Step 1.2.4

Combine and simplify the denominator.

Tap for more steps...

Step 1.2.4.1

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

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{\sqrt{1 + x}}{2 \sqrt{1 + x} \sqrt{1 + x}}$

Step 1.2.4.2

Move $\sqrt{1 + x}$ .

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{\sqrt{1 + x}}{2 (\sqrt{1 + x} \sqrt{1 + x})}$

Step 1.2.4.3

Raise $\sqrt{1 + x}$ to the power of $1$ .

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{\sqrt{1 + x}}{2 ({\sqrt{1 + x}}^{1} \sqrt{1 + x})}$

Step 1.2.4.4

Raise $\sqrt{1 + x}$ to the power of $1$ .

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{\sqrt{1 + x}}{2 ({\sqrt{1 + x}}^{1} {\sqrt{1 + x}}^{1})}$

Step 1.2.4.5

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

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{\sqrt{1 + x}}{2 {\sqrt{1 + x}}^{1 + 1}}$

Step 1.2.4.6

Add $1$ and $1$ .

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{\sqrt{1 + x}}{2 {\sqrt{1 + x}}^{2}}$

Step 1.2.4.7

Rewrite ${\sqrt{1 + x}}^{2}$ as $1 + x$ .

Tap for more steps...

Step 1.2.4.7.1

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

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{\sqrt{1 + x}}{2 {({(1 + x)}^{\frac{1}{2}})}^{2}}$

Step 1.2.4.7.2

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

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{\sqrt{1 + x}}{2 {(1 + x)}^{\frac{1}{2} \cdot 2}}$

Step 1.2.4.7.3

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

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{\sqrt{1 + x}}{2 {(1 + x)}^{\frac{2}{2}}}$

Step 1.2.4.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.4.7.4.1

Cancel the common factor.

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{\sqrt{1 + x}}{2 {(1 + x)}^{\frac{2}{2}}}$

Step 1.2.4.7.4.2

Rewrite the expression.

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{\sqrt{1 + x}}{2 {(1 + x)}^{1}}$

Step 1.2.4.7.5

Simplify.

$\frac{1}{{(2 - y)}^{2}} \frac{d y}{d x} = \frac{\sqrt{1 + x}}{2 (1 + x)}$

Step 1.3

Rewrite the equation.

$\frac{1}{{(2 - y)}^{2}} d y = \frac{\sqrt{1 + x}}{2 (1 + x)} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Let $u_{1} = 2 - y$ . Then $d u_{1} = - d y$ , so $- d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 2.2.1.1

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

Tap for more steps...

Step 2.2.1.1.1

Rewrite.

$\frac{- 1}{1}$

Step 2.2.1.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 2.2.1.2

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

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

Step 2.2.2

Move the negative in front of the fraction.

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

Step 2.2.3

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

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

Step 2.2.4

Apply basic rules of exponents.

Tap for more steps...

Step 2.2.4.1

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

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

Step 2.2.4.2

Multiply the exponents in ${({u_{1}}^{2})}^{- 1}$ .

Tap for more steps...

Step 2.2.4.2.1

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

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

Step 2.2.4.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.5

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

$- (- {u_{1}}^{- 1} + C_{1}) = \int \frac{\sqrt{1 + x}}{2 (1 + x)} d x$

Step 2.2.6

Simplify.

Tap for more steps...

Step 2.2.6.1

Rewrite $- (- {u_{1}}^{- 1} + C_{1})$ as $- - \frac{1}{u_{1}} + C_{1}$ .

$- - \frac{1}{u_{1}} + C_{1} = \int \frac{\sqrt{1 + x}}{2 (1 + x)} d x$

Step 2.2.6.2

Simplify.

Tap for more steps...

Step 2.2.6.2.1

Multiply $- 1$ by $- 1$ .

$1 \frac{1}{u_{1}} + C_{1} = \int \frac{\sqrt{1 + x}}{2 (1 + x)} d x$

Step 2.2.6.2.2

Multiply $\frac{1}{u_{1}}$ by $1$ .

$\frac{1}{u_{1}} + C_{1} = \int \frac{\sqrt{1 + x}}{2 (1 + x)} d x$

Step 2.2.7

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

$\frac{1}{2 - y} + C_{1} = \int \frac{\sqrt{1 + x}}{2 (1 + x)} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

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

$\frac{1}{2 - y} + C_{1} = \frac{1}{2} \int \frac{\sqrt{1 + x}}{1 + x} d x$

Step 2.3.2

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

Tap for more steps...

Step 2.3.2.1

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

Tap for more steps...

Step 2.3.2.1.1

Differentiate $1 + x$ .

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

Step 2.3.2.1.2

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

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

Step 2.3.2.1.3

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

$0 + \frac{d}{d x} [x]$

Step 2.3.2.1.4

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

$0 + 1$

Step 2.3.2.1.5

Add $0$ and $1$ .

$1$

Step 2.3.2.2

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

$\frac{1}{2 - y} + C_{1} = \frac{1}{2} \int \frac{\sqrt{u_{2}}}{u_{2}} d u_{2}$

Step 2.3.3

Simplify the expression.

Tap for more steps...

Step 2.3.3.1

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

$\frac{1}{2 - y} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{\frac{1}{2}}}{u_{2}} d u_{2}$

Step 2.3.3.2

Simplify.

Tap for more steps...

Step 2.3.3.2.1

Move ${u_{2}}^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{1}{2 - y} + C_{1} = \frac{1}{2} \int \frac{1}{u_{2} \cdot {u_{2}}^{- \frac{1}{2}}} d u_{2}$

Step 2.3.3.2.2

Multiply $u_{2}$ by ${u_{2}}^{- \frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 2.3.3.2.2.1

Multiply $u_{2}$ by ${u_{2}}^{- \frac{1}{2}}$ .

Tap for more steps...

Step 2.3.3.2.2.1.1

Raise $u_{2}$ to the power of $1$ .

$\frac{1}{2 - y} + C_{1} = \frac{1}{2} \int \frac{1}{{u_{2}}^{1} {u_{2}}^{- \frac{1}{2}}} d u_{2}$

Step 2.3.3.2.2.1.2

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

$\frac{1}{2 - y} + C_{1} = \frac{1}{2} \int \frac{1}{{u_{2}}^{1 - \frac{1}{2}}} d u_{2}$

Step 2.3.3.2.2.2

Write $1$ as a fraction with a common denominator.

$\frac{1}{2 - y} + C_{1} = \frac{1}{2} \int \frac{1}{{u_{2}}^{\frac{2}{2} - \frac{1}{2}}} d u_{2}$

Step 2.3.3.2.2.3

Combine the numerators over the common denominator.

$\frac{1}{2 - y} + C_{1} = \frac{1}{2} \int \frac{1}{{u_{2}}^{\frac{2 - 1}{2}}} d u_{2}$

Step 2.3.3.2.2.4

Subtract $1$ from $2$ .

$\frac{1}{2 - y} + C_{1} = \frac{1}{2} \int \frac{1}{{u_{2}}^{\frac{1}{2}}} d u_{2}$

Step 2.3.3.3

Apply basic rules of exponents.

Tap for more steps...

Step 2.3.3.3.1

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

$\frac{1}{2 - y} + C_{1} = \frac{1}{2} \int {({u_{2}}^{\frac{1}{2}})}^{- 1} d u_{2}$

Step 2.3.3.3.2

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

Tap for more steps...

Step 2.3.3.3.2.1

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

$\frac{1}{2 - y} + C_{1} = \frac{1}{2} \int {u_{2}}^{\frac{1}{2} \cdot - 1} d u_{2}$

Step 2.3.3.3.2.2

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

$\frac{1}{2 - y} + C_{1} = \frac{1}{2} \int {u_{2}}^{\frac{- 1}{2}} d u_{2}$

Step 2.3.3.3.2.3

Move the negative in front of the fraction.

$\frac{1}{2 - y} + C_{1} = \frac{1}{2} \int {u_{2}}^{- \frac{1}{2}} d u_{2}$

Step 2.3.4

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

$\frac{1}{2 - y} + C_{1} = \frac{1}{2} (2 {u_{2}}^{\frac{1}{2}} + C_{2})$

Step 2.3.5

Simplify.

Tap for more steps...

Step 2.3.5.1

Rewrite $\frac{1}{2} (2 {u_{2}}^{\frac{1}{2}} + C_{2})$ as $\frac{1}{2} \cdot 2 {u_{2}}^{\frac{1}{2}} + C_{2}$ .

$\frac{1}{2 - y} + C_{1} = \frac{1}{2} \cdot 2 {u_{2}}^{\frac{1}{2}} + C_{2}$

Step 2.3.5.2

Simplify.

Tap for more steps...

Step 2.3.5.2.1

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

$\frac{1}{2 - y} + C_{1} = \frac{2}{2} {u_{2}}^{\frac{1}{2}} + C_{2}$

Step 2.3.5.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.5.2.2.1

Cancel the common factor.

$\frac{1}{2 - y} + C_{1} = \frac{2}{2} {u_{2}}^{\frac{1}{2}} + C_{2}$

Step 2.3.5.2.2.2

Rewrite the expression.

$\frac{1}{2 - y} + C_{1} = 1 {u_{2}}^{\frac{1}{2}} + C_{2}$

Step 2.3.5.2.3

Multiply ${u_{2}}^{\frac{1}{2}}$ by $1$ .

$\frac{1}{2 - y} + C_{1} = {u_{2}}^{\frac{1}{2}} + C_{2}$

Step 2.3.6

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

$\frac{1}{2 - y} + C_{1} = {(1 + x)}^{\frac{1}{2}} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$\frac{1}{2 - y} = {(1 + x)}^{\frac{1}{2}} + K$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 - y, 1, 1$

Step 3.1.2

Remove parentheses.

$2 - y, 1, 1$

Step 3.1.3

The LCM of one and any expression is the expression.

$2 - y$

Step 3.2

Multiply each term in $\frac{1}{2 - y} = {(1 + x)}^{\frac{1}{2}} + K$ by $2 - y$ to eliminate the fractions.

Tap for more steps...

Step 3.2.1

Multiply each term in $\frac{1}{2 - y} = {(1 + x)}^{\frac{1}{2}} + K$ by $2 - y$ .

$\frac{1}{2 - y} (2 - y) = {(1 + x)}^{\frac{1}{2}} (2 - y) + K (2 - y)$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $2 - y$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{1}{2 - y} (2 - y) = {(1 + x)}^{\frac{1}{2}} (2 - y) + K (2 - y)$

Step 3.2.2.1.2

Rewrite the expression.

$1 = {(1 + x)}^{\frac{1}{2}} (2 - y) + K (2 - y)$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Simplify each term.

Tap for more steps...

Step 3.2.3.1.1

Apply the distributive property.

$1 = {(1 + x)}^{\frac{1}{2}} \cdot 2 + {(1 + x)}^{\frac{1}{2}} (- y) + K (2 - y)$

Step 3.2.3.1.2

Move $2$ to the left of ${(1 + x)}^{\frac{1}{2}}$ .

$1 = 2 \cdot {(1 + x)}^{\frac{1}{2}} + {(1 + x)}^{\frac{1}{2}} (- y) + K (2 - y)$

Step 3.2.3.1.3

Rewrite using the commutative property of multiplication.

$1 = 2 \cdot {(1 + x)}^{\frac{1}{2}} - {(1 + x)}^{\frac{1}{2}} y + K (2 - y)$

Step 3.2.3.1.4

Apply the distributive property.

$1 = 2 {(1 + x)}^{\frac{1}{2}} - {(1 + x)}^{\frac{1}{2}} y + K \cdot 2 + K (- y)$

Step 3.2.3.1.5

Move $2$ to the left of $K$ .

$1 = 2 {(1 + x)}^{\frac{1}{2}} - {(1 + x)}^{\frac{1}{2}} y + 2 \cdot K + K (- y)$

Step 3.2.3.1.6

Rewrite using the commutative property of multiplication.

$1 = 2 {(1 + x)}^{\frac{1}{2}} - {(1 + x)}^{\frac{1}{2}} y + 2 K - K y$

Step 3.2.3.2

Reorder factors in $2 {(1 + x)}^{\frac{1}{2}} - {(1 + x)}^{\frac{1}{2}} y + 2 K - K y$ .

$1 = 2 {(1 + x)}^{\frac{1}{2}} - y {(1 + x)}^{\frac{1}{2}} + 2 K - K y$

Step 3.3

Solve the equation.

Tap for more steps...

Step 3.3.1

Rewrite the equation as $2 {(1 + x)}^{\frac{1}{2}} - y {(1 + x)}^{\frac{1}{2}} + 2 K - K y = 1$ .

$2 {(1 + x)}^{\frac{1}{2}} - y {(1 + x)}^{\frac{1}{2}} + 2 K - K y = 1$

Step 3.3.2

Move all terms not containing $y$ to the right side of the equation.

Tap for more steps...

Step 3.3.2.1

Subtract $2 {(1 + x)}^{\frac{1}{2}}$ from both sides of the equation.

$- y {(1 + x)}^{\frac{1}{2}} + 2 K - K y = 1 - 2 {(1 + x)}^{\frac{1}{2}}$

Step 3.3.2.2

Subtract $2 K$ from both sides of the equation.

$- y {(1 + x)}^{\frac{1}{2}} - K y = 1 - 2 {(1 + x)}^{\frac{1}{2}} - 2 K$

Step 3.3.3

Factor $y$ out of $- y {(1 + x)}^{\frac{1}{2}} - K y$ .

Tap for more steps...

Step 3.3.3.1

Factor $y$ out of $- y {(1 + x)}^{\frac{1}{2}}$ .

$y (- 1 {(1 + x)}^{\frac{1}{2}}) - K y = 1 - 2 {(1 + x)}^{\frac{1}{2}} - 2 K$

Step 3.3.3.2

Factor $y$ out of $- K y$ .

$y (- 1 {(1 + x)}^{\frac{1}{2}}) + y (- K) = 1 - 2 {(1 + x)}^{\frac{1}{2}} - 2 K$

Step 3.3.3.3

Factor $y$ out of $y (- 1 {(1 + x)}^{\frac{1}{2}}) + y (- K)$ .

$y (- 1 {(1 + x)}^{\frac{1}{2}} - K) = 1 - 2 {(1 + x)}^{\frac{1}{2}} - 2 K$

Step 3.3.4

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

$y (- {(1 + x)}^{\frac{1}{2}} - K) = 1 - 2 {(1 + x)}^{\frac{1}{2}} - 2 K$

Step 3.3.5

Divide each term in $y (- {(1 + x)}^{\frac{1}{2}} - K) = 1 - 2 {(1 + x)}^{\frac{1}{2}} - 2 K$ by $- {(1 + x)}^{\frac{1}{2}} - K$ and simplify.

Tap for more steps...

Step 3.3.5.1

Divide each term in $y (- {(1 + x)}^{\frac{1}{2}} - K) = 1 - 2 {(1 + x)}^{\frac{1}{2}} - 2 K$ by $- {(1 + x)}^{\frac{1}{2}} - K$ .

$\frac{y (- {(1 + x)}^{\frac{1}{2}} - K)}{- {(1 + x)}^{\frac{1}{2}} - K} = \frac{1}{- {(1 + x)}^{\frac{1}{2}} - K} + \frac{- 2 {(1 + x)}^{\frac{1}{2}}}{- {(1 + x)}^{\frac{1}{2}} - K} + \frac{- 2 K}{- {(1 + x)}^{\frac{1}{2}} - K}$

Step 3.3.5.2

Simplify the left side.

Tap for more steps...

Step 3.3.5.2.1

Divide $y$ by $1$ .

$y = \frac{1}{- {(1 + x)}^{\frac{1}{2}} - K} + \frac{- 2 {(1 + x)}^{\frac{1}{2}}}{- {(1 + x)}^{\frac{1}{2}} - K} + \frac{- 2 K}{- {(1 + x)}^{\frac{1}{2}} - K}$

Step 3.3.5.3

Simplify the right side.

Tap for more steps...

Step 3.3.5.3.1

Simplify each term.

Tap for more steps...

Step 3.3.5.3.1.1

Move the negative in front of the fraction.

$y = \frac{1}{- {(1 + x)}^{\frac{1}{2}} - K} - \frac{2 {(1 + x)}^{\frac{1}{2}}}{- {(1 + x)}^{\frac{1}{2}} - K} + \frac{- 2 K}{- {(1 + x)}^{\frac{1}{2}} - K}$

Step 3.3.5.3.1.2

Move the negative in front of the fraction.

$y = \frac{1}{- {(1 + x)}^{\frac{1}{2}} - K} - \frac{2 {(1 + x)}^{\frac{1}{2}}}{- {(1 + x)}^{\frac{1}{2}} - K} - \frac{2 K}{- {(1 + x)}^{\frac{1}{2}} - K}$

Step 3.3.5.3.2

Combine the numerators over the common denominator.

$y = \frac{1 - 2 {(1 + x)}^{\frac{1}{2}}}{- {(1 + x)}^{\frac{1}{2}} - K} - \frac{2 K}{- {(1 + x)}^{\frac{1}{2}} - K}$

Step 4

Simplify the constant of integration.

$y = \frac{1 - 2 {(1 + x)}^{\frac{1}{2}}}{- {(1 + x)}^{\frac{1}{2}} + K} - \frac{K}{- {(1 + x)}^{\frac{1}{2}} + K}$