Let be a differentiable function for which and .

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Let

g

be a differentiable function for which

\dfrac{dg}{dx}=\dfrac{1}{1+x^{2}}+2x

and

g(0)=-1

. (a) Determine the explicit expression for

g(x)

that satisfies the given condition. (b) Compute

g(1)

, showing all steps in your work. (c) Describe the end behavior of

g(x)

x\to\infty

, supporting your conclusion algebraically.