• Title: Integration by Parts

  • Series: Real Analysis

  • Chapter: Riemann Integral

  • YouTube-Title: Real Analysis 58 | Integration by Parts

  • Bright video: https://youtu.be/2EH3XnaDPKU

  • Dark video: https://youtu.be/5g0QKwT8CPE

  • Quiz: Test your knowledge

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  • Quiz Content

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

    A1: $\int_a^b f^\prime(x) g(x) dx = \int_a^b g(x) f^\prime(x) dx$

    A2: $\int_a^b f^\prime(x) g(x) dx = f(x) g(x)|_a^b - \int_a^b g(x) f^\prime(x) dx$

    A3: $\int_a^b f^\prime(x) g(x) dx = f(x) g(x)|_a^b - \int_a^b f(x) g^\prime(x) dx$

    A4: $\int_a^b f^\prime(x) g(x) dx = f(x) g(x)|_a^b + \int_a^b f^\prime(x) g(x) dx$

    A5: $\int_a^b f^\prime(x) g(x) dx = f^\prime(x) g(x)|_a^b - \int_a^b f^\prime(x) g(x) dx$

    Q2: What is an antiderivative of the function $f:\mathbb{R} \rightarrow \mathbb{R}$ given by $$f(t) = t \sin(t)$$

    A1: $$ -t \cos(t) - \sin(t) $$

    A2: $$ -t \cos(t) + \sin(t) $$

    A3: $$ t \cos(t) - \sin(t) $$

    A4: $$ -t \cos(t) + \sin(t) $$

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

    A1: $0$

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

    A3: $1$

    A4: $\frac{1}{2}$

    A5: $e$

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