Evaluate the integral below and sketch the graph of the family member that passes through the point (1, 2).

Question

Evaluate the integral below and sketch the graph of the family member that passes through the point (1, 2). ​  ​

Quizplus · Accepted Answer

First, we need to evaluate the integral.

$y=\int \frac{2x}{x^2+1}dx=\ln(x^2+1)+C$

We know that the family member passes through the point (1, 2), so we can substitute these values into the equation above to solve for C:

$2=\ln(1^2+1)+C$
$2=\ln2+C$
$C=2-\ln2$

Therefore, the equation of the family member passing through (1, 2) is:

$y=\ln(x^2+1)+2-\ln2$

Now, we can sketch the graph by plotting a few points:

When x=0, y=$\ln1+2-\ln2=2-\ln2$
When x=1, y=$\ln2+2-\ln2=2$
When x=2, y=$\ln5+2-\ln2\approx 2.7$
When x=-1, y=$\ln2+2-\ln2=2$
When x=-2, y=$\ln5+2-\ln2\approx 2.7$

We can see that the graph is symmetric about the y-axis, and approaches but never touches the x-axis. Therefore, the best choice is B.

Evaluate the Integral Below and Sketch the Graph of the Family