Problem 41 - Essential Calculus

Question

Use a computer algebra system to find or evaluate the integral. $$ \int \frac{1}{1+\sqrt{x}} d x $$

Step-by-Step Solution

Answer

The integral of \(\frac{1}{1+\sqrt{x}} dx\) is \(2 \ln | \frac{1}{\sqrt{x}} + 1| + C\).

Step 1: Perform the substitution

Let \(u = \sqrt{x}\). This means that \(x = u^2\). In terms of \(u\), \(dx\) becomes \(2udu\). Substituting these into the integral gives us: \(\int \frac{1}{1+u} * 2u du\).

Step 2: Simplify the integral

The integral can be simplified by multiplying through the \(2u\): \(\int \frac{2u}{1+u} du\). This can be further simplified into a form that's easier to integrate by dividing each term in the numerator by \(u\). This gives: \(\int \frac{2}{\frac{1}{u}+1} du\).

Step 3: Evaluate the integral

Now, we can straightforwardly integrate this function: \(\int \frac{2}{\frac{1}{u}+1} du = 2 \int \frac{1}{\frac{1}{u}+1} du = 2 \ln |\frac{1}{u} + 1| + C\). We've accounted for the constant of integration \(C\).

Step 4: Convert the answer back to terms of \(x\)

Finally, we need to convert the answer back to terms of \(x\) by substituting \(u = \sqrt{x}\) back into the answer: \(2 \ln | \frac{1}{\sqrt{x}} + 1| + C\).