Calculus Examples

$\frac{d y}{d x} = \frac{x + 1}{\sqrt{x}}$ , $y (1) = 0$

Step 1

Rewrite the equation.

$d y = \frac{x + 1}{\sqrt{x}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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Step 2.3.3.1

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

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

Step 2.3.3.2

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

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

Step 2.3.3.3

Move the negative in front of the fraction.

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

Step 2.3.4

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

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Step 2.3.4.1

Apply the distributive property.

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

Step 2.3.4.2

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

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

Step 2.3.4.3

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

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

Step 2.3.4.4

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

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

Step 2.3.4.5

Combine the numerators over the common denominator.

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

Step 2.3.4.6

Subtract $1$ from $2$ .

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

Step 2.3.4.7

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

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

Step 2.3.5

Split the single integral into multiple integrals.

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

Step 2.3.6

By the Power Rule, the integral of $x^{\frac{1}{2}}$ with respect to $x$ is $\frac{2}{3} x^{\frac{3}{2}}$ .

$y + C_{1} = \frac{2}{3} x^{\frac{3}{2}} + C_{2} + \int x^{- \frac{1}{2}} d x$

Step 2.3.7

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

$y + C_{1} = \frac{2}{3} x^{\frac{3}{2}} + C_{2} + 2 x^{\frac{1}{2}} + C_{3}$

Step 2.3.8

Simplify.

$y + C_{1} = \frac{2}{3} x^{\frac{3}{2}} + 2 x^{\frac{1}{2}} + C_{4}$

Step 2.4

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

$y = \frac{2}{3} x^{\frac{3}{2}} + 2 x^{\frac{1}{2}} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $1$ for $x$ and $0$ for $y$ in $y = \frac{2}{3} x^{\frac{3}{2}} + 2 x^{\frac{1}{2}} + K$ .

$0 = \frac{2}{3} \cdot 1^{\frac{3}{2}} + 2 \cdot 1^{\frac{1}{2}} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\frac{2}{3} \cdot 1^{\frac{3}{2}} + 2 \cdot 1^{\frac{1}{2}} + K = 0$ .

$\frac{2}{3} \cdot 1^{\frac{3}{2}} + 2 \cdot 1^{\frac{1}{2}} + K = 0$

Step 4.2

Simplify $\frac{2}{3} \cdot 1^{\frac{3}{2}} + 2 \cdot 1^{\frac{1}{2}} + K$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

One to any power is one.

$\frac{2}{3} \cdot 1 + 2 \cdot 1^{\frac{1}{2}} + K = 0$

Step 4.2.1.2

Multiply $\frac{2}{3}$ by $1$ .

$\frac{2}{3} + 2 \cdot 1^{\frac{1}{2}} + K = 0$

Step 4.2.1.3

One to any power is one.

$\frac{2}{3} + 2 \cdot 1 + K = 0$

Step 4.2.1.4

Multiply $2$ by $1$ .

$\frac{2}{3} + 2 + K = 0$

Step 4.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2}{3} + 2 \cdot \frac{3}{3} + K = 0$

Step 4.2.3

Combine $2$ and $\frac{3}{3}$ .

$\frac{2}{3} + \frac{2 \cdot 3}{3} + K = 0$

Step 4.2.4

Combine the numerators over the common denominator.

$\frac{2 + 2 \cdot 3}{3} + K = 0$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $2$ by $3$ .

$\frac{2 + 6}{3} + K = 0$

Step 4.2.5.2

Add $2$ and $6$ .

$\frac{8}{3} + K = 0$

Step 4.3

Subtract $\frac{8}{3}$ from both sides of the equation.

$K = - \frac{8}{3}$

Step 5

Substitute $- \frac{8}{3}$ for $K$ in $y = \frac{2}{3} x^{\frac{3}{2}} + 2 x^{\frac{1}{2}} + K$ and simplify.

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Step 5.1

Substitute $- \frac{8}{3}$ for $K$ .

$y = \frac{2}{3} x^{\frac{3}{2}} + 2 x^{\frac{1}{2}} - \frac{8}{3}$

Step 5.2

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

$y = \frac{2 x^{\frac{3}{2}}}{3} + 2 x^{\frac{1}{2}} - \frac{8}{3}$