Calculus Examples

$f (x) = 5 x^{\frac{1}{4}} - 7 x^{\frac{3}{4}}$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

$F (x) = \int 5 x^{\frac{1}{4}} - 7 x^{\frac{3}{4}} d x$

Step 3

Split the single integral into multiple integrals.

$\int 5 x^{\frac{1}{4}} d x + \int - 7 x^{\frac{3}{4}} d x$

Step 4

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int x^{\frac{1}{4}} d x + \int - 7 x^{\frac{3}{4}} d x$

Step 5

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

$5 (\frac{4}{5} x^{\frac{5}{4}} + C) + \int - 7 x^{\frac{3}{4}} d x$

Step 6

Since $- 7$ is constant with respect to $x$ , move $- 7$ out of the integral.

$5 (\frac{4}{5} x^{\frac{5}{4}} + C) - 7 \int x^{\frac{3}{4}} d x$

Step 7

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

$5 (\frac{4}{5} x^{\frac{5}{4}} + C) - 7 (\frac{4}{7} x^{\frac{7}{4}} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$5 (\frac{4}{5}) x^{\frac{5}{4}} - 7 (\frac{4}{7}) x^{\frac{7}{4}} + C$

Step 8.2

Simplify.

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Step 8.2.1

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

$\frac{5 \cdot 4}{5} x^{\frac{5}{4}} - 7 (\frac{4}{7}) x^{\frac{7}{4}} + C$

Step 8.2.2

Multiply $5$ by $4$ .

$\frac{20}{5} x^{\frac{5}{4}} - 7 (\frac{4}{7}) x^{\frac{7}{4}} + C$

Step 8.2.3

Cancel the common factor of $20$ and $5$ .

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Step 8.2.3.1

Factor $5$ out of $20$ .

$\frac{5 \cdot 4}{5} x^{\frac{5}{4}} - 7 (\frac{4}{7}) x^{\frac{7}{4}} + C$

Step 8.2.3.2

Cancel the common factors.

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Step 8.2.3.2.1

Factor $5$ out of $5$ .

$\frac{5 \cdot 4}{5 (1)} x^{\frac{5}{4}} - 7 (\frac{4}{7}) x^{\frac{7}{4}} + C$

Step 8.2.3.2.2

Cancel the common factor.

$\frac{5 \cdot 4}{5 \cdot 1} x^{\frac{5}{4}} - 7 (\frac{4}{7}) x^{\frac{7}{4}} + C$

Step 8.2.3.2.3

Rewrite the expression.

$\frac{4}{1} x^{\frac{5}{4}} - 7 (\frac{4}{7}) x^{\frac{7}{4}} + C$

Step 8.2.3.2.4

Divide $4$ by $1$ .

$4 x^{\frac{5}{4}} - 7 (\frac{4}{7}) x^{\frac{7}{4}} + C$

Step 8.2.4

Combine $- 7$ and $\frac{4}{7}$ .

$4 x^{\frac{5}{4}} + \frac{- 7 \cdot 4}{7} x^{\frac{7}{4}} + C$

Step 8.2.5

Multiply $- 7$ by $4$ .

$4 x^{\frac{5}{4}} + \frac{- 28}{7} x^{\frac{7}{4}} + C$

Step 8.2.6

Cancel the common factor of $- 28$ and $7$ .

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Step 8.2.6.1

Factor $7$ out of $- 28$ .

$4 x^{\frac{5}{4}} + \frac{7 \cdot - 4}{7} x^{\frac{7}{4}} + C$

Step 8.2.6.2

Cancel the common factors.

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Step 8.2.6.2.1

Factor $7$ out of $7$ .

$4 x^{\frac{5}{4}} + \frac{7 \cdot - 4}{7 (1)} x^{\frac{7}{4}} + C$

Step 8.2.6.2.2

Cancel the common factor.

$4 x^{\frac{5}{4}} + \frac{7 \cdot - 4}{7 \cdot 1} x^{\frac{7}{4}} + C$

Step 8.2.6.2.3

Rewrite the expression.

$4 x^{\frac{5}{4}} + \frac{- 4}{1} x^{\frac{7}{4}} + C$

Step 8.2.6.2.4

Divide $- 4$ by $1$ .

$4 x^{\frac{5}{4}} - 4 x^{\frac{7}{4}} + C$

Step 9

The answer is the antiderivative of the function $f (x) = 5 x^{\frac{1}{4}} - 7 x^{\frac{3}{4}}$ .

$F (x) =$ $4 x^{\frac{5}{4}} - 4 x^{\frac{7}{4}} + C$