Solve the problem by integration.
-The general expression for the slope of a curve is $\frac { 4 x + 4 } { x ^ { 2 } + 4 x }$. Find the equation of the curve if it passes through (1, 0 ).
A) $y = \ln \left| \frac { x ( x - 4 ) ^ { 3 } } { 125 } \right|$
B) $y = \ln \left| \frac { x ( x + 4 ) ^ { 3 } } { 5 } \right|$
C) $y = \ln \left| x ( x + 4 ) ^ { 3 } \right|$
D) $y = \ln \left| \frac { x ( x + 4 ) ^ { 3 } } { 125 } \right|$

Question

Quizplus · Accepted Answer

We start by integrating the given expression for the slope:

$$\int \frac{4x+4}{x^2+4x} dx = \int \frac{4(x+1)}{x(x+4)} dx$$
$$= \int \left(\frac{1}{x} - \frac{1}{x+4}\right) 4 dx = 4\ln\left|\frac{x}{x+4}\right| + C$$

Using the fact that the curve passes through (1,0), we can find the value of C:

$$0 = 4\ln\left|\frac{1}{1+4}\right| + C = -4\ln 5 + C \implies C = 4\ln 5$$

So the equation of the curve is:

$$y = \int \frac{4x+4}{x^2+4x} dx + C = 4\ln\left|\frac{x}{x+4}\right| + 4\ln 5$$
$$= \ln\left|\frac{x}{x+4}\right|^4 + \ln 5^4 = \ln\left|\frac{x(x+4)^3}{125}\right|$$

Therefore, the best choice is D.

Solve the Problem by Integration 4x+4x2+4x\frac { 4 x + 4 } { x ^ { 2 } + 4 x }x2+4x4x+4​

Solve the Problem by Integration $\frac { 4 x + 4 } { x ^ { 2 } + 4 x }$