• 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

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: ra57_sub_eng.srt missing

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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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