Math / Calculus

Question $\int \frac{(1+\sqrt[3]{x})^{2}}{\sqrt[3]{x}} d x$

Studdy Solution

STEP 1

1. The integral $\int \frac{(1+\sqrt[3]{x})^{2}}{\sqrt[3]{x}} d x$ involves an algebraic function with a cube root.
2. The integrand can be simplified using algebraic manipulations and substitution.
3. The goal is to find the antiderivative in terms of $x$ .

STEP 2

1. Simplify the integrand.
2. Use substitution to facilitate integration.
3. Integrate the simplified function.
4. Back-substitute to return to the original variable, if necessary.

STEP 3

Rewrite the integrand $\frac{(1+\sqrt[3]{x})^{2}}{\sqrt[3]{x}}$ in a simpler form by expressing $\sqrt[3]{x}$ as $x^{1/3}$ .
$\frac{(1+x^{1/3})^{2}}{x^{1/3}}$

STEP 4

Expand the numerator $(1 + x^{1/3})^2$ .
$(1 + x^{1/3})^2 = 1 + 2x^{1/3} + x^{2/3}$

STEP 5

Substitute the expanded form back into the integrand.
$\int \frac{1 + 2x^{1/3} + x^{2/3}}{x^{1/3}} dx$

STEP 6

Simplify the integrand by dividing each term in the numerator by $x^{1/3}$ .
$\int \left( x^{-1/3} + 2 + x^{1/3} \right) dx$

STEP 7

Integrate each term separately.
$\int x^{-1/3} dx + \int 2 dx + \int x^{1/3} dx$

STEP 8

Use the power rule for integration, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ , for each term.
$\int x^{-1/3} dx = \frac{x^{2/3}}{2/3} + C_1 = \frac{3}{2} x^{2/3} + C_1$ $\int 2 dx = 2x + C_2$ $\int x^{1/3} dx = \frac{x^{4/3}}{4/3} + C_3 = \frac{3}{4} x^{4/3} + C_3$

STEP 9

Combine the results of the integrations, combining constants into a single constant $C$ .
$\frac{3}{2} x^{2/3} + 2x + \frac{3}{4} x^{4/3} + C$

STEP 10

Since we have already returned to the original variable $x$ , this step confirms the final answer.
The antiderivative is:
$\frac{3}{2} x^{2/3} + 2x + \frac{3}{4} x^{4/3} + C$
Solution: $\frac{3}{2} x^{2/3} + 2x + \frac{3}{4} x^{4/3} + C$

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