Q1: Let $f,\phi: \mathbb{R} \rightarrow \mathbb{R}$ be two continuously differentiable functions. What is the correct substitution rule?

A1: $\int_a^b f(x) dx = \int_{\phi(a)}^{\phi(b)} f(t) dt $

A2: $\int_{\phi(a)}^{\phi(b)} f(x) dx = \int_{\phi(a)}^{\phi(b)} f(t) dt $

A3: $\int_{\phi(a)}^{\phi(b)} f(x) dx = \int_{a}^{b} f(t) \phi^\prime(t) dt $

A4: $\int_{\phi(a)}^{\phi(b)} f(x) dx = \int_{a}^{b} f(\phi(t)) \phi^\prime(t) dt $

A5: $\int_{\phi(a)}^{\phi(b)} f(x) dx = \int_{a}^{b} f(\phi(t)) dt $

Q2: What is the antiderivative of the function $f:\mathbb{R} \rightarrow \mathbb{R}$ given by $$f(t) = t^4 \cos(t^5)$$

A1: $$ \sin(t) $$

A2: $$ -\frac{1}{5} \sin(t^5) $$

A3: $$ \frac{1}{5} \cos(t^5) $$

A4: $$ \frac{1}{5} \sin(t^5) $$

Q3: What is the integral $\int_0^1 t \exp(t^2) , dt$?

A1: $0$

A2: $\frac{1}{2} e - 1$

A3: $\frac{1}{2} (e - 1)$

A4: $\frac{1}{2} (e - 2)$

A5: $\frac{1}{3} (e - 1)$