Question

c) \\( \frac { d } { d x } \left( \int _ { 2 } ^ { x ^ { 3 } } \arctan u d u \
ight) = \\)

Explanation:

Step1: Apply the Fundamental Theorem of Calculus and Chain Rule

Let \(F(u)\) be an antiderivative of \(\arctan u\), i.e., \(F^\prime(u)=\arctan u\). Then \(\int_{2}^{x^{3}}\arctan u\ du=F(x^{3}) - F(2)\).
Differentiating with respect to \(x\) gives \(\frac{d}{dx}(F(x^{3})-F(2))\).
By the chain rule, \(\frac{d}{dx}F(x^{3})=F^\prime(x^{3})\cdot\frac{d}{dx}(x^{3})\) and \(\frac{d}{dx}F(2) = 0\) (since \(F(2)\) is a constant).

Step2: Substitute \(F^\prime(u)=\arctan u\) and find \(\frac{d}{dx}(x^{3})\)

Since \(F^\prime(u)=\arctan u\), then \(F^\prime(x^{3})=\arctan(x^{3})\).
And \(\frac{d}{dx}(x^{3}) = 3x^{2}\).

Answer:

\(3x^{2}\arctan(x^{3})\)