# Evaluate the integral by using a substitution prior to integration by parts. \int x^2\sqrt{x +...

## Question:

Evaluate the integral by using a substitution prior to integration by parts.

{eq}\int x^2\sqrt{x + 25}dx \\ a. \ \frac{\left ( 30x^2 - 600x + 400 \right )\left ( x + 25 \right )^{\frac{3}{2}}}{105} + C \\ b. \ \frac{\left ( 30x^2 - 600x + 10,000 \right )\left ( x + 25 \right )^{\frac{3}{2}}}{105} + C \\ c. \ \frac{\left ( 30x^2 - 600x + 10,000 \right )\sqrt{\left ( x + 25 \right )}}{105} + C \\ d. \ \frac{\left ( 15x^2 - 300x + 5,000 \right )\left ( x + 25 \right )^{\frac{3}{2}}}{105} + C {/eq}

## Integrals:

The given problem on indefinite integrals is typical in nature. Here, we will first make use of the substitution method, in order to simplify our given function and thereafter proceed with the integration.

## Answer and Explanation:

{eq}\int x^2\sqrt{x + 25} \ dx {/eq}

Put,

{eq}\begin{align*} \ & (x+25) = t \end{align*} {/eq}

Differentiating the above, we get:

{eq}\begin{align*} \ & dx = dt \end{align*} {/eq}

Substituting the above values, we get:

{eq}\begin{align*} \ & = \int (t-25)^2 \sqrt{t} \ dt \\ \\ \ & = \int (t^2+625-50t) \sqrt{t} \ dt \\ \\ \ & = \int (t^{5/2}+625t^{1/2}-50t^{3/2}) \ dt \\ \\ \ & = \left[ \frac{t^{7/2}}{\frac{7}{2}}+\frac{625t^{3/2}}{\frac{3}{2}}-\frac{50t^{5/2}}{\frac{5}{2}} \right] +c \\ \\ \ & = \frac{2t^{7/2}}{7}+\frac{2 \cdot 625t^{3/2}}{3}-\frac{100t^{5/2}}{5} +c \\ \\ \ & = t^{3/2} (\frac{2t^2}{7}+\frac{1250}{3}- \frac{100t}{5} ) +c \end{align*} {/eq}

Substituting the value of {eq}t {/eq} back, we get:

{eq}\begin{align*} \ & = (x+25)^{3/2} \left(\frac{2(x+25)^2}{7}+\frac{1250}{3}-\frac{100(x+25)}{5} \right) +c \\ \\ \ & = (x+25)^{3/2} \left(\frac{2(x^2+625+50x)}{7}+\frac{1250}{3}-\frac{100x+2500}{5} \right) +c \\ \\ \ & = (x+25)^{3/2} \left(\frac{15(2x^2+1250+100x) + 35 \cdot 1250-21(100x+2500)}{105} \right) +c \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left[ \mathrm{ Least \ common \ multiple \ (l.c.m)} \right] \\ \\ \ & = (x+25)^{3/2} \left(\frac{30x^2+18750+1500x + 43750 -2100x - 52500}{105} \right) +c \\ \\ \ & = (x+25)^{3/2} \left(\frac{30x^2-600x +10000}{105} \right) +c \end{align*} {/eq}