Calculus Examples

$\int (\frac{1}{7} - \frac{1}{y^{\frac{5}{4}}}) d y$

Step 1

Remove parentheses.

$\int \frac{1}{7} - \frac{1}{y^{\frac{5}{4}}} d y$

Step 2

Split the single integral into multiple integrals.

$\int \frac{1}{7} d y + \int - \frac{1}{y^{\frac{5}{4}}} d y$

Step 3

Apply the constant rule.

$\frac{1}{7} y + C + \int - \frac{1}{y^{\frac{5}{4}}} d y$

Step 4

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

$\frac{1}{7} y + C - \int \frac{1}{y^{\frac{5}{4}}} d y$

Step 5

Apply basic rules of exponents.

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Step 5.1

Move $y^{\frac{5}{4}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{7} y + C - \int {(y^{\frac{5}{4}})}^{- 1} d y$

Step 5.2

Multiply the exponents in ${(y^{\frac{5}{4}})}^{- 1}$ .

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Step 5.2.1

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

$\frac{1}{7} y + C - \int y^{\frac{5}{4} \cdot - 1} d y$

Step 5.2.2

Multiply $\frac{5}{4} \cdot - 1$ .

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Step 5.2.2.1

Combine $\frac{5}{4}$ and $- 1$ .

$\frac{1}{7} y + C - \int y^{\frac{5 \cdot - 1}{4}} d y$

Step 5.2.2.2

Multiply $5$ by $- 1$ .

$\frac{1}{7} y + C - \int y^{\frac{- 5}{4}} d y$

Step 5.2.3

Move the negative in front of the fraction.

$\frac{1}{7} y + C - \int y^{- \frac{5}{4}} d y$

Step 6

By the Power Rule, the integral of $y^{- \frac{5}{4}}$ with respect to $y$ is $- 4 y^{- \frac{1}{4}}$ .

$\frac{1}{7} y + C - (- 4 y^{- \frac{1}{4}} + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

$\frac{1}{7} y - (- 4 y^{- \frac{1}{4}}) + C$

Step 7.2

Multiply $- 4$ by $- 1$ .

$\frac{1}{7} y + 4 y^{- \frac{1}{4}} + C$