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Title: Integration by Parts
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Series: Real Analysis
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Chapter: Riemann Integral
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YouTube-Title: Real Analysis 58 | Integration by Parts
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Quiz: Test your knowledge
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Subtitle on GitHub: ra58_sub_eng.srt missing
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Timestamps (n/a)
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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: $ \displaystyle \int_a^b f^\prime(x) g(x) dx $ $ \displaystyle = \int_a^b g(x) f^\prime(x) dx$
A2: $ \displaystyle \int_a^b f^\prime(x) g(x) dx $ $ \displaystyle = f(x) g(x)\Big|_a^b $ $ \displaystyle - \int_a^b g(x) f^\prime(x) dx$
A3: $ \displaystyle \int_a^b f^\prime(x) g(x) dx $ $ \displaystyle = f(x) g(x)\Big|_a^b $ $ \displaystyle - \int_a^b f(x) g^\prime(x) dx$
A4: $ \displaystyle \int_a^b f^\prime(x) g(x) dx $ $ \displaystyle = f(x) g(x)\Big|_a^b $ $ \displaystyle + \int_a^b f^\prime(x) g(x) dx$
A5: $ \displaystyle \int_a^b f^\prime(x) g(x) dx $ $ \displaystyle = f^\prime(x) g(x)\Big|_a^b $ $ \displaystyle - \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 $$f(t) = t \sin(t) \text{ ?}$$
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 $\displaystyle \int_0^1 t \exp(t) , dt \text{ ?}$
A1: $0$
A2: $\frac{1}{2} e - 1$
A3: $1$
A4: $\frac{1}{2}$
A5: $e$
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Last update: 2025-09
