Evaluate the integral if it exists. $$\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$$
A) $1 - 10 \ln x + C$
B)$x - 10 \ln x - \frac { 25 } { x } + C$
C) $\ln x - 25 x + C$
D)$x - \frac { 1 } { 10 \ln x } + C$
E) none of these

Question

Quizplus · Accepted Answer

Expanding the square term gives:
$$\int\left(\frac{25-10x+x^2}{x^2}\right)dx$$
Splitting this into two integrals:
$$\int\frac{25}{x^2}dx-10\int\frac{1}{x}dx+\int\frac{x^2}{x^2}dx$$
Solving each integral separately:
$$\int\frac{25}{x^2}dx=-\frac{25}{x}+C_1$$
$$10\int\frac{1}{x}dx=-10\ln|x|+C_2$$
$$\int\frac{x^2}{x^2}dx=\int1dx=x+C_3$$
Putting everything back together:
$$\int\left(\frac{ 5 - x } { x } \right) ^ { 2 } d x = \int \frac{25}{x^2}-10\cdot\frac{1}{x}+\frac{x^2}{x^2}dx= \left(-\frac{25}{x}-10\ln|x|+x\right)+C$$
Simplifying yields the final answer of $$\boxed{\textbf{(B) } x - 10 \ln x - \frac { 25 } { x } + C}$$

Evaluate the Integral If It Exists ∫(5−xx)2dx\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x∫(x5−x​)2dx

Evaluate the Integral If It Exists $\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$