Calculus Examples

$f (x) = 7 x^{\frac{2}{5}} + 4 x^{- \frac{4}{5}}$

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.

$F (x) = \int 7 x^{\frac{2}{5}} + 4 x^{- \frac{4}{5}} d x$

Step 3

Split the single integral into multiple integrals.

$\int 7 x^{\frac{2}{5}} d x + \int 4 x^{- \frac{4}{5}} d x$

Step 4

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

$7 \int x^{\frac{2}{5}} d x + \int 4 x^{- \frac{4}{5}} d x$

Step 5

By the Power Rule, the integral of $x^{\frac{2}{5}}$ with respect to $x$ is $\frac{5}{7} x^{\frac{7}{5}}$ .

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

Step 6

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 7

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

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

Step 8

Simplify.

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Simplify.

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

Simplify.

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Combine $7$ and $\frac{5}{7}$ .

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

Multiply $7$ by $5$ .

$\frac{35}{7} x^{\frac{7}{5}} + 4 \cdot 5 x^{\frac{1}{5}} + C$

Cancel the common factor of $35$ and $7$ .

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Factor $7$ out of $35$ .

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

Cancel the common factors.

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Factor $7$ out of $7$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $5$ by $1$ .

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

Multiply $4$ by $5$ .

$5 x^{\frac{7}{5}} + 20 x^{\frac{1}{5}} + C$

Step 9

The answer is the antiderivative of the function $f (x) = 7 x^{\frac{2}{5}} + 4 x^{- \frac{4}{5}}$ .

$F (x) =$ $5 x^{\frac{7}{5}} + 20 x^{\frac{1}{5}} + C$