#C05 · Calculus & Analysis · Lv.3View in 3D Mind-Map

Integration by Substitution & by Parts

Brook Taylor & Johann Bernoulli · 1715 CE

\int u\,dv = uv - \int v\,du,\quad \int f(g(x))g'(x)\,dx = \int f(u)\,du

(uv)' = u'v + uv' \implies uv' = (uv)' - u'v

Reverses the chain rule (u-substitution) and product rule (integration by parts) to integrate products of functions, enabling Fourier transforms, Gamma functions, and FEM.

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Upper limit x

\int_0^{2} t e^t\,dt = \Big[(t-1)e^t\Big]_0^{2} = (1)e^{2} + 1 = 8.3891

정적분 ∫₀^2 t·e^t dt

8.3891

부정적분 (x-1)e^x

7.3891

피적분함수 x·e^x

14.7781

Prerequisite Path (11 STEP)

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Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 2#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 3#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 4#A06Logarithm Laws & Change of Base

\log_a(xy) = \log_a x + \log_a y

STEP 5#A08Binomial Theorem & Pascal's Triangle

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

STEP 6#C01Limits, Continuity & Euler's Number e

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828

STEP 7#C02Derivative Definition & Instantaneous Rate

f'(x) = \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

STEP 8#A07Arithmetic & Geometric Series Sum

S_n = \frac{a(1 - r^n)}{1 - r}

STEP 9#C03Product Rule & Chain Rule (Backpropagation Core)

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

STEP 10#C04Fundamental Theorem of Calculus (FTC)

\int_a^b f(x)\

STEP 11#C05Integration by Substitution & by Parts

\int u\

🚀 Next Unlocks (Derived Formulas) (1)

#C09Fourier Series & Fourier Transform (FFT)

Geometric & Intuitive Insight

가로 u, 세로 v인 직사각형 넓이 uv의 변화량은 ∫ u dv + ∫ v du 와 같습니다. 따라서 미분하기 쉬운 함수 u와 적분하기 쉬운 함수 dv로 나누면 어려운 적분 ∫ u dv 를 쉬운 적분 ∫ v du 로 바꿀 수 있습니다.

Step-by-Step Derivation

Step 1.1단계: 곱의 미분법 d(uv) = u dv + v du 를 세웁니다.
Step 2.2단계: 이항하여 정리하면 u dv = d(uv) - v du 가 됩니다.
Step 3.3단계: 양변을 적분하면 ∫ u dv = uv - ∫ v du 부분적분 공식이 완성됩니다.

AI & Coding Real-World Use

Finite Element Method (FEM weak formulation) physics simulation and Laplace/Fourier derivative rules

#FEM Physics Sim#Fourier Transform Proof#Signal Laplace

const intXeX = (x: number) => (x - 1) * Math.exp(x) + 1; // ∫_0^x t*e^t dt

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