Calculus Examples

$\frac{d y}{d x} = \frac{2 x}{\sqrt{2 x^{2} - 1}}$ , $y (5) = 4$

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.2

Let $u = 2 x^{2} - 1$ . Then $d u = 4 x d x$ , so $\frac{1}{4} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

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

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Step 2.3.2.1.1

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

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

Step 2.3.2.1.2

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

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

Step 2.3.2.1.3

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

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Step 2.3.2.1.3.1

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

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

Step 2.3.2.1.3.2

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

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

Step 2.3.2.1.3.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [- 1]$

Step 2.3.2.1.4

Differentiate using the Constant Rule.

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Step 2.3.2.1.4.1

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

$4 x + 0$

Step 2.3.2.1.4.2

Add $4 x$ and $0$ .

$4 x$

Step 2.3.2.2

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

$y + C_{1} = 2 \int \frac{1}{\sqrt{u}} \cdot \frac{1}{4} d u$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{4}$ .

$y + C_{1} = 2 \int \frac{1}{\sqrt{u} \cdot 4} d u$

Step 2.3.3.2

Move $4$ to the left of $\sqrt{u}$ .

$y + C_{1} = 2 \int \frac{1}{4 \sqrt{u}} d u$

Step 2.3.4

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$y + C_{1} = 2 (\frac{1}{4} \int \frac{1}{\sqrt{u}} d u)$

Step 2.3.5

Simplify the expression.

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Step 2.3.5.1

Simplify.

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Step 2.3.5.1.1

Combine $\frac{1}{4}$ and $2$ .

$y + C_{1} = \frac{2}{4} \int \frac{1}{\sqrt{u}} d u$

Step 2.3.5.1.2

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

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Step 2.3.5.1.2.1

Factor $2$ out of $2$ .

$y + C_{1} = \frac{2 (1)}{4} \int \frac{1}{\sqrt{u}} d u$

Step 2.3.5.1.2.2

Cancel the common factors.

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Step 2.3.5.1.2.2.1

Factor $2$ out of $4$ .

$y + C_{1} = \frac{2 \cdot 1}{2 \cdot 2} \int \frac{1}{\sqrt{u}} d u$

Step 2.3.5.1.2.2.2

Cancel the common factor.

$y + C_{1} = \frac{2 \cdot 1}{2 \cdot 2} \int \frac{1}{\sqrt{u}} d u$

Step 2.3.5.1.2.2.3

Rewrite the expression.

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

Step 2.3.5.2

Apply basic rules of exponents.

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Step 2.3.5.2.1

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

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

Step 2.3.5.2.2

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

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

Step 2.3.5.2.3

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

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Step 2.3.5.2.3.1

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

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

Step 2.3.5.2.3.2

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

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

Step 2.3.5.2.3.3

Move the negative in front of the fraction.

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

Step 2.3.6

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

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

Step 2.3.7

Simplify.

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Step 2.3.7.1

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

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

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

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

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

Step 2.3.7.2.2

Cancel the common factor of $2$ .

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Step 2.3.7.2.2.1

Cancel the common factor.

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

Step 2.3.7.2.2.2

Rewrite the expression.

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

Step 2.3.7.2.3

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

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

Step 2.3.8

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

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

Step 2.4

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

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

Step 3

Use the initial condition to find the value of $K$ by substituting $5$ for $x$ and $4$ for $y$ in $y = {(2 x^{2} - 1)}^{\frac{1}{2}} + K$ .

$4 = {(2 \cdot 5^{2} - 1)}^{\frac{1}{2}} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as ${(2 \cdot 5^{2} - 1)}^{\frac{1}{2}} + K = 4$ .

${(2 \cdot 5^{2} - 1)}^{\frac{1}{2}} + K = 4$

Step 4.2

Simplify each term.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raise $5$ to the power of $2$ .

${(2 \cdot 25 - 1)}^{\frac{1}{2}} + K = 4$

Step 4.2.1.2

Multiply $2$ by $25$ .

${(50 - 1)}^{\frac{1}{2}} + K = 4$

Step 4.2.2

Subtract $1$ from $50$ .

$49^{\frac{1}{2}} + K = 4$

Step 4.2.3

Rewrite $49$ as $7^{2}$ .

${(7^{2})}^{\frac{1}{2}} + K = 4$

Step 4.2.4

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

$7^{2 (\frac{1}{2})} + K = 4$

Step 4.2.5

Cancel the common factor of $2$ .

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Step 4.2.5.1

Cancel the common factor.

$7^{2 (\frac{1}{2})} + K = 4$

Step 4.2.5.2

Rewrite the expression.

$7^{1} + K = 4$

Step 4.2.6

Evaluate the exponent.

$7 + K = 4$

Step 4.3

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

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Step 4.3.1

Subtract $7$ from both sides of the equation.

$K = 4 - 7$

Step 4.3.2

Subtract $7$ from $4$ .

$K = - 3$

Step 5

Substitute $- 3$ for $K$ in $y = {(2 x^{2} - 1)}^{\frac{1}{2}} + K$ and simplify.

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Step 5.1

Substitute $- 3$ for $K$ .

$y = {(2 x^{2} - 1)}^{\frac{1}{2}} - 3$