Integrals

Integrals determine the cumulative of something.

\color{#48cae4}
position = \int_{t_0}^{t_f} velocity⠀ ∂time + P_₀

Simplification

{\color{tomato}k} \int_{\color{tomato}ka}^{\color{tomato}kb} s(x) dx =  \int_{a}^{b} s({\color{tomato}k}x) dx

Integrales inmediatas

\int x^{\color{gold}n} = \frac{x^{\color{gold}n+1}}{\color{gold}n+1}  {\color{tomato} + 𝙲}

\int {\color{gold}\frac{1}{x}} = \int {\color{gold}\ddot{x}} = {\color{gold} 𝙻𝚗\left | x \right | } {\color{tomato} + 𝙲}

\int {\color{gold}℮^x} = {\color{gold}℮^x} {\color{tomato} + 𝙲}

\int {\color{gold}𝒂^x} = {\color{gold} \frac{𝒂^x}{𝙻𝚗𝒂}} {\color{tomato} + 𝙲}

\int {\color{gold}cos𝑥} = {\color{gold} sin𝑥} {\color{tomato} + 𝙲}

\int {\color{gold}sin𝑥} = {\color{gold} -cos𝑥} {\color{tomato} + 𝙲}

\int \frac{1}{1+x^2} = arctan(𝑥)   {\color{tomato} + 𝙲}

\int \frac{1}{\sqrt{1+x^2}} = arcsin(𝑥)  {\color{tomato} + 𝙲}

\int \frac{1}{cos^2x} = tan(𝑥)   {\color{tomato} + 𝙲}

\int \frac{1}{sin^2x} = -cotan(𝑥)   {\color{tomato} + 𝙲}

\int \frac{1}{\sqrt{1+x^2}} = 𝙻𝚗(𝑥 + \sqrt{1+x^2})   {\color{tomato} + 𝙲}

\int \frac{1}{\sqrt{𝑥²-1}} = 𝙻𝚗(𝑥+ \sqrt{𝑥²-1})   {\color{tomato} + 𝙲}

∫𝑓(𝑔(𝑥))𝑔'(𝑥)d𝑥 = 𝐹(𝑔(𝑥)) \text{, si } ∫𝑓(𝑥)d𝑥 = 𝐹(𝑥)

∫𝑢d𝑣 = 𝑢𝑣 - ∫𝑣d𝑢

𝑢=𝑓  d𝑣= 𝑔 d𝑥

Inversal Integration

\int_0^p f(x) \, dx = {\color{orange}p f(p)} 
{\color{gold} - \int_0^p f^{-1}(f(x)) f'(x) \, dx }

∫^p f(x) = {\color{orange} p·f(p)} 
{\color{gold} - \int^{f(p)} f^{-1}(f) df(x) }

This formula breaks down as follows:

1. 🟠 $( p f(p) )$ The rectangle approximation.
2. 🟡 $( \int_0^p f^{-1}(f(x)) f'(x) , dx )$ : The correction term that accounts for the curvature of $( f(x) )$ .