Calculus Examples

$\int (x^{\frac{3}{4}} - \frac{1}{x^{\frac{3}{4}}}) d x$

Step 1

Remove parentheses.

$\int x^{\frac{3}{4}} - \frac{1}{x^{\frac{3}{4}}} d x$

Step 2

Split the single integral into multiple integrals.

$\int x^{\frac{3}{4}} d x + \int - \frac{1}{x^{\frac{3}{4}}} d x$

Step 3

By the Power Rule, the integral of $x^{\frac{3}{4}}$ with respect to $x$ is $\frac{4}{7} x^{\frac{7}{4}}$ .

$\frac{4}{7} x^{\frac{7}{4}} + C + \int - \frac{1}{x^{\frac{3}{4}}} d x$

Step 4

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

$\frac{4}{7} x^{\frac{7}{4}} + C - \int \frac{1}{x^{\frac{3}{4}}} d x$

Step 5

Apply basic rules of exponents.

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Step 5.1

Move $x^{\frac{3}{4}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{4}{7} x^{\frac{7}{4}} + C - \int {(x^{\frac{3}{4}})}^{- 1} d x$

Step 5.2

Multiply the exponents in ${(x^{\frac{3}{4}})}^{- 1}$ .

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Step 5.2.1

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

$\frac{4}{7} x^{\frac{7}{4}} + C - \int x^{\frac{3}{4} \cdot - 1} d x$

Step 5.2.2

Multiply $\frac{3}{4} \cdot - 1$ .

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Step 5.2.2.1

Combine $\frac{3}{4}$ and $- 1$ .

$\frac{4}{7} x^{\frac{7}{4}} + C - \int x^{\frac{3 \cdot - 1}{4}} d x$

Step 5.2.2.2

Multiply $3$ by $- 1$ .

$\frac{4}{7} x^{\frac{7}{4}} + C - \int x^{\frac{- 3}{4}} d x$

Step 5.2.3

Move the negative in front of the fraction.

$\frac{4}{7} x^{\frac{7}{4}} + C - \int x^{- \frac{3}{4}} d x$

Step 6

By the Power Rule, the integral of $x^{- \frac{3}{4}}$ with respect to $x$ is $4 x^{\frac{1}{4}}$ .

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

Step 7

Simplify.

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Step 7.1

Simplify.

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

Step 7.2

Multiply $- 1$ by $4$ .

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