Problem 12 - Essential Calculus

Question

Find the indefinite integral. $$ \int \frac{1}{x^{2 / 3}\left(1+x^{1 / 3}\right)} d x $$

Step-by-Step Solution

Answer

The integral of \( \int \frac{1}{x^{2 / 3}(1+x^{1 / 3})} dx \) is \( 3 \ln |1+ x^{1 / 3}|+ C \)

Step 1: Identify Substitution

Let's take \( u = x^{1 / 3} \). Then, \(\frac{du}{dx} = \frac{1}{3}x^{-2 / 3} \). Then, \( dx = 3u^2 du \). This will simplify the denominator and make our integral easier to solve.

Step 2: Substitute into Integral

Now, substitute in the variable u and the new dx into the original integral: \(\int \frac{1}{u^2 (1+u)} 3u^2 du \). This simplifies to \(3 \int \frac{1}{1+u} du \).

Step 3: Integrate

Now, we can easily compute this integral, which results to \(3 \ln |1+u| + C \). This is the result in terms of u.

Step 4: Substitute Back

In the last step, we need to substitute back to our original variable x. Since we have set \( u = x^{1 / 3} \), the final result is hence \(3 \ln |1+ x^{1 / 3}|+ C \).