Question

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

Short answer

The final result of the definite integral after performing the calculation in the last step.

Step-by-step solution

Perform the Substitution

Since the denominator is \(\sqrt{5 + 2x}\), we make a substitution for this part, let \(u = 5 + 2x\). Hence, the differential \(du = 2dx\). Next, change the limits of the integral according to the substitution. When \(x = 0\), \(u = 5\), and when \(x = 5\), \(u = 15\).

Change the Integral According to the Substitution

Because \(du = 2dx\) and \(x = (u-5)/2\), the integral can be rewritten in terms of \(u\) as follows: \[\frac{1}{2}\int_{5}^{15} \frac{u-5}{\sqrt{u}} du \], Here, the factor of \(1/2\) is resulted due to conversion \(2dx = du\).

Simplify and Compute the Integral

Breaking the integral, it is written as follows: \[\frac{1}{2}\int_{5}^{15}( u^{1/2} - 5u^{-1/2} ) du\] Now, we can find the antiderivative of this expression, yielding: \[\frac{1}{2}[\frac{2}{3}u^{3/2} - 10u^{1/2}]^5_{15}\], and then evaluate this between the new limits.

Evaluate the Definite Integral

Finally, substitute the limits back into the antiderivative function: \[\frac{1}{2}[\frac{2}{3}15^{3/2} - 10*15^{1/2}]-\frac{1}{2}[\frac{2}{3}*5^{3/2} - 10*5^{1/2}]\]Calculate the numerical result for the final answer.