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\noindent This problem is intended to be used right after problem w2Dv2.

\begin{questions}

\question % workshop bank Ch. 5 -> esergel55
The goal of this problem is to calculate $\frac{d}{dx} \int_{2x}^{x^2} \ln(t) \, dt$ using the Fundamental Theorem of Calculus, Part II (FTCII) and then to double-check this calculation by using the Fundamental Theorem of Calculus, Part I (FTCI). Note that this second method is rarely possible: usually an FTCII-type question will involve an integral that you do not know how to calculate. You must be able to use FTCII as in parts (a)-(c) below.

\begin{parts}

\part Use FTCII to compute $\frac{d}{dx} \int_{1}^{x^2} \ln(t) \, dt$.
% 2x\ln(x^2) = 4x\ln(x)

\part Use FTCII to compute $\frac{d}{dx} \int_{1}^{2x} \ln(t) \, dt$.
% 2\ln(2x) = 4\ln(x) + 2\ln(2)

\part Use parts (a) and (b) to compute $\frac{d}{dx} \int_{2x}^{x^2} \ln(t) \, dt$.
% 2x\ln(x^2) - 2\ln(2x)

\part Use your answer from problem w2Dv2 part (d) together with FTCI to compute $\int_{2x}^{x^2} \ln(t) \, dt$.
% int \ln(t) \, dt = t\ln(t) - t + C
% int_{2x}^{x^2} \ln(t) \, dt = [ x^2 \ln(x^2) - x^2 ] - [ 2x ln(2x) - 2x ]
% = x^2 \ln(x^2) - x^2 - 2x ln(2x) + 2x

\part Take the derivative of your answer in part (d) to verify your answer from part (c).
% derivative = 2x \ln(x^2) + 2x - 2x - 2ln(2x) - 2 + 2
% = 2x \ln(x^2) - 2\ln(2x)

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