Problem 16 - Essential Calculus

Question

Find the indefinite integral by \(u\) -substitution. (Hint: Let \(u\) be the denominator of the integrand.) $$ \int \frac{1}{1+\sqrt{3 x}} d x $$

Step-by-Step Solution

Answer

The indefinite integral of the function is \( \frac{2}{3} (1 + \sqrt{3x}) + C \).

Step 1: Identify and Set up the u-substitution

First, identify the denominator of the integral expression that would become more manageable with a substitution. In this exercise, the term inside the square root function (3x) is a suitable choice. If \( u = 1 + \sqrt{3x} \), we then take derivative of \( u \) to find \( du \). The derivative of \( u \) with respect to \( x \) is \( du = \frac{3}{2\sqrt{3x}} dx \).

Step 2: Rearrange for dx

Rearrange the derivative equation to solve for \( dx \): \( dx = \frac{2}{3} u du \). This gives us a form to directly replace \( dx \) with \( du \) in our integral.

Step 3: Substitute in the Integral

Now we substitute \( u \) for \( 1 + \sqrt{3x} \) and \( dx \) for \( \frac{2}{3} u du \) in our original integral. The new integral becomes \( \int \frac{1}{u} \cdot \frac{2}{3} u du = \frac{2}{3} \int du \).

Step 4: Integrate and Back-substitute u

The integral \( \frac{2}{3} \int du \) simplifies to \( \frac{2}{3} u + C \), where \( C \) is the constant of integration. Finally, replace \( u \) with the original expression \( 1 + \sqrt{3x} \) to get \( \frac{2}{3} (1 + \sqrt{3x}) + C \).