Calculus Examples

$f (x) = 9 x^{\frac{4}{3}} + 4 x^{\frac{5}{4}}$

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.

$F (x) = \int 9 x^{\frac{4}{3}} + 4 x^{\frac{5}{4}} d x$

Step 3

Split the single integral into multiple integrals.

$\int 9 x^{\frac{4}{3}} d x + \int 4 x^{\frac{5}{4}} d x$

Step 4

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int x^{\frac{4}{3}} d x + \int 4 x^{\frac{5}{4}} d x$

Step 5

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

$9 (\frac{3}{7} x^{\frac{7}{3}} + C) + \int 4 x^{\frac{5}{4}} d x$

Step 6

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

$9 (\frac{3}{7} x^{\frac{7}{3}} + C) + 4 \int x^{\frac{5}{4}} d x$

Step 7

By the Power Rule, the integral of $x^{\frac{5}{4}}$ with respect to $x$ is $\frac{4}{9} x^{\frac{9}{4}}$ .

$9 (\frac{3}{7} x^{\frac{7}{3}} + C) + 4 (\frac{4}{9} x^{\frac{9}{4}} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$9 (\frac{3}{7}) x^{\frac{7}{3}} + 4 (\frac{4}{9}) x^{\frac{9}{4}} + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $9$ and $\frac{3}{7}$ .

$\frac{9 \cdot 3}{7} x^{\frac{7}{3}} + 4 (\frac{4}{9}) x^{\frac{9}{4}} + C$

Step 8.2.2

Multiply $9$ by $3$ .

$\frac{27}{7} x^{\frac{7}{3}} + 4 (\frac{4}{9}) x^{\frac{9}{4}} + C$

Step 8.2.3

Combine $4$ and $\frac{4}{9}$ .

$\frac{27}{7} x^{\frac{7}{3}} + \frac{4 \cdot 4}{9} x^{\frac{9}{4}} + C$

Step 8.2.4

Multiply $4$ by $4$ .

$\frac{27}{7} x^{\frac{7}{3}} + \frac{16}{9} x^{\frac{9}{4}} + C$

Step 9

The answer is the antiderivative of the function $f (x) = 9 x^{\frac{4}{3}} + 4 x^{\frac{5}{4}}$ .

$F (x) =$ $\frac{27}{7} x^{\frac{7}{3}} + \frac{16}{9} x^{\frac{9}{4}} + C$