Calculus Examples

$\frac{d c}{d y} = 0.5 + \frac{0.2}{\sqrt{y}}$

Step 1

Rewrite the equation.

$d c = (0.5 + \frac{0.2}{\sqrt{y}}) d y$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d c = \int 0.5 + \frac{0.2}{\sqrt{y}} d y$

Step 2.2

Apply the constant rule.

$c + C_{1} = \int 0.5 + \frac{0.2}{\sqrt{y}} d y$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$c + C_{1} = \int 0.5 d y + \int \frac{0.2}{\sqrt{y}} d y$

Step 2.3.2

Apply the constant rule.

$c + C_{1} = 0.5 y + C_{2} + \int \frac{0.2}{\sqrt{y}} d y$

Step 2.3.3

Since $0.2$ is constant with respect to $y$ , move $0.2$ out of the integral.

$c + C_{1} = 0.5 y + C_{2} + 0.2 \int \frac{1}{\sqrt{y}} d y$

Step 2.3.4

Apply basic rules of exponents.

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Step 2.3.4.1

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

$c + C_{1} = 0.5 y + C_{2} + 0.2 \int \frac{1}{y^{\frac{1}{2}}} d y$

Step 2.3.4.2

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

$c + C_{1} = 0.5 y + C_{2} + 0.2 \int {(y^{\frac{1}{2}})}^{- 1} d y$

Step 2.3.4.3

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

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Step 2.3.4.3.1

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

$c + C_{1} = 0.5 y + C_{2} + 0.2 \int y^{\frac{1}{2} \cdot - 1} d y$

Step 2.3.4.3.2

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

$c + C_{1} = 0.5 y + C_{2} + 0.2 \int y^{\frac{- 1}{2}} d y$

Step 2.3.4.3.3

Move the negative in front of the fraction.

$c + C_{1} = 0.5 y + C_{2} + 0.2 \int y^{- \frac{1}{2}} d y$

Step 2.3.5

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

$c + C_{1} = 0.5 y + C_{2} + 0.2 (2 y^{\frac{1}{2}} + C_{3})$

Step 2.3.6

Simplify.

$c + C_{1} = 0.5 y + 0.4 y^{\frac{1}{2}} + C_{4}$

Step 2.4

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

$c = 0.5 y + 0.4 y^{\frac{1}{2}} + K$