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Title: Integration by Substitution
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Series: Real Analysis
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Chapter: Riemann Integral
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YouTube-Title: Real Analysis 57 | Integration by Substitution
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Bright video: https://youtu.be/wmZCoV6Y0_c
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Dark video: https://youtu.be/O1_pWnSQiwU
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ra57_sub_eng.srt missing
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Timestamps (n/a)
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Subtitle in English (n/a)
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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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Last update: 2024-10
