Question

Evaluate the definite integral. $$ \int_{0}^{5} \frac{x}{(4+x)^{2}} d x $$

Short answer

Your short answer is \(5 + 4*ln(\frac{4}{9})\)

Step-by-step solution

Identify the substitution

Identify a suitable substitution. Notice the term \((4+x)^{2}\) in the denominator - it would make sense to use this for a substitute. Let \(u = (4+x)\). Then, calculate \(du\) which would be \(du = dx\)

Change the limits of the integration

Substitute the limits of integration from x-values to u-values. When \(x = 0\), \(u = (4+0) = 4\), and when \(x = 5\), \(u = (4+5) = 9\)

Substitute x and dx in the integral formula

Express \(x\) in terms of \(u\) and substitute this as well as \(dx\) in the integral. From \(u = 4 + x\), you can express \(x = u - 4\). Substituting this yields, \(\int_{4}^{9} \frac{u-4}{u^{2}} du\)

Simplify the Integrals

Break down this integral into simpler parts for easy evaluation. It simplifies to \(\int_{4}^{9} (1 - \frac{4}{u}) du= \int_{4}^{9} du - 4*\int_{4}^{9} \frac{1}{u} du\

Evaluate the Integrals

Evaluate each simpler integral separately. The integral \(\int du\) is just \(u\), and the integral of \(\int \frac{1}{u} du\) is \(ln(u)\). Thus this step yields \(u-4*ln(u)|_{4}^{9} = [9-4*ln(9)] - [4-4*ln(4)]\

Simplify the Expression

Calculate the expression obtained from step 5: \(9-4*ln(9) - 4 + 4*ln(4) = 5 - 4*ln(9) + 4*ln(4) = 5 + 4*ln(\frac{4}{9})\