Problem 17 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

The indefinite integral of the given function is \( 2(\sqrt{x} - 3) + 6 \cdot ln|\sqrt{x} - 3| + C \).

Step 1: Identifying the Substitution Variable 'u'

Let \( u = \sqrt{x} - 3\). Now, we will also need to compute the differential \( du \) to substitute \( dx \) in the integral. This is obtained by differentiating \( u \) with respect to \( x \).

Step 2: Finding the Differential 'du'

Differentiate \( u \) with respect to \( x \) to get \( du \) which equals \( \frac{1}{2\sqrt{x}} dx \). We will need to multiply this by \( 2\sqrt{x} \) on both sides to be able to substitute for \( dx \) in the integral. This gives \( du \cdot 2\sqrt{x} = dx \).

Step 3: Substituting 'u' and 'du' in the Original Integral

We rewrite the integrand using \( u \) and \( du \). Our integral now becomes \( \int \frac{u+3}{u} \cdot du \cdot 2(u+3) \).

Step 4: Simplifying the Integral and Solve

The integral can now be simplified and solved yielding a sum of two integrals \( 2\int du + 6 \int u^{-1} \cdot du \). Solving this yields \( 2u + 6 \cdot ln|u| + C \).

Step 5: Substituting 'u' back with Original Expression

Finally, as the last step, we substitute 'u' back with its original expression. So, \( 2u + 6 \cdot ln|u| + C \) becomes \( 2(\sqrt{x} - 3) + 6 \cdot ln|\sqrt{x} - 3| + C \).