• Title: Integration by Substitution

• Series: Real Analysis

• Chapter: Riemann Integral

• YouTube-Title: Real Analysis 57 | Integration by Substitution

• Bright video: https://youtu.be/wmZCoV6Y0_c

• Dark video: https://youtu.be/O1_pWnSQiwU

• Subtitle on GitHub: ra57_sub_eng.srt missing

• Timestamps (n/a)
• Subtitle in English (n/a)
• Quiz Content

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

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