"Evaluate the following indefinite integral using integration by parts:
*integral sign* tan^-1(x) dx"
I let u = tan^-1(x) and dv = dx. Is that right?
Oops! I just found out that I don't need to know how to do this type of question.
To evaluate the given indefinite integral using integration by parts, you need to select the appropriate choices for u and dv.
You've correctly chosen u = tan^(-1)(x) and dv = dx. However, it's better to choose dv as the entire function without ignoring the dx. In this case, you can choose dv = dx.
Now, let's proceed with the integration by parts formula:
∫ u dv = uv - ∫ v du
To apply this formula, we need to find du and v.
Differentiating both sides of the equation u = tan^(-1)(x), we get:
du = d(tan^(-1)(x))
To simplify this, we'll make use of the identity: d(tan^(-1)(x)) = (1 / (1 + x^2)) dx
Therefore, du = (1 / (1 + x^2)) dx.
To find v, we integrate dv = dx:
v = ∫ dv = ∫ dx = x
Now, we have all the necessary components to use the integration by parts formula:
∫ tan^(-1)(x) dx = uv - ∫ v du
= (x)(tan^(-1)(x)) - ∫ x (1 / (1 + x^2)) dx
Now, simplify the integral on the right side and solve it. Finally, you will have your answer.