what is the integral of x/(1+x^2)^2 dx This is a question of a past AP exam of calculus BC

Question

what is the integral of x/(1+x^2)^2 dx This is a question of a past AP exam of calculus BC

i know i have to use the substitution method where u=(1+x^2), du=2xdx. I have to find the integral but i'm just focused on the coefficient because i get a different answer for that part

then you set the integral to .5(int)1/u^2 du,

but my answer choice on the AP exam test says that instead of 1/2 as the coefficient, it's 1/4 and i don't see where the 4 comes from.

Answer 1

To find the integral of x/(1+x^2)^2 dx, you correctly started by using the substitution method.

Let u = 1 + x^2, then du = 2x dx.

Now, let's rearrange the equation to solve for x dx. Dividing both sides of du = 2x dx by 2x, we get dx = du / (2x).

Substituting this expression for dx in terms of du and x into the original integral, we have:

∫ x / (1 + x^2)^2 dx = ∫ [x / (1 + x^2)^2] (du / 2x)

Next, we can simplify the expression by canceling out the x terms:

∫ [1 / (1 + x^2)] (du / 2)

Now, we have the integral in terms of u:

∫ 1 / (2(1 + x^2)) du.

To write this integral in terms of u, we can further simplify the expression:

∫ 1 / (2u) du = (1/2) ∫ 1 / u du

Here, we can integrate 1/u with respect to u:

(1/2) ∫ 1 / u du = (1/2) ln |u| + C

Note: C represents the constant of integration.

Since u = 1 + x^2, we get:

(1/2) ln |1 + x^2| + C

Now, to address the discrepancy in the coefficient, you mentioned that your answer choice on the AP exam has 1/4 as the coefficient instead of 1/2.

It's possible that the answer choices are expressed in different forms, such as a different constant of integration or an equivalent expression. It's important to carefully review the answer choices and compare them to the expression you obtained.

Alternatively, it's also possible that there might have been a mistake or typo in the answer choices. If you are confident in your solution and have followed the correct steps, it's worth bringing this discrepancy to the attention of your teacher or the exam administrator for clarification.

Remember that it's always important to carefully show your work and follow the given instructions on the exam to ensure that you are getting the correct answer.