Aprende con Inteligencia

Calcular la siguiente integral indefinida:
$$ \int \frac{\sin x}{\sin 3x} \, dx $$

(a) $\frac{1}{\sqrt{3}} \log \left| \frac{\sqrt{3} + \tan x}{\sqrt{3} - \tan x} \right| + c$
(b) $\frac{1}{2\sqrt{3}} \log \left| \frac{\sqrt{3} + \tan x}{\sqrt{3} - \tan x} \right| + c$
(c) $\log \left| \frac{\sqrt{3} + \tan x}{\sqrt{3} - \tan x} \right| + c$
(d) $\frac{1}{\sqrt{3}} \log \left| \frac{\sqrt{3} + \tan x}{\sqrt{3} \tan x} \right| + c$

Calcule la integral definida:
$$\int_0^{\pi/2} \frac{dx}{1+\sin(x)}$$

Evaluar:
$$ \int (x + \sqrt{x^{2}+1})^{10} dx $$

Calcular la integral:
$$ \int \left( \frac{1}{x} \log\left(\frac{x}{e^x}\right) \right) \, dx $$

(a) $\frac{1}{2} e^x - \ln x + c$ \\
(b) $\frac{1}{2} \ln x - e^x + c$ \\
(c) $\frac{1}{2} \ln^2 x - x + c$ \\
(d) None.

Resuelva la integral:
$$\int \sqrt{x \sqrt{x \sqrt{x \dots}}} \, dx$$

Evaluar la integral:
$$ \int \frac{\cos x - \sin x + 1 - e^x}{e^x + \sin x + x} dx $$

Evaluar la parte entera de la expresión:
$$ \left\lfloor 10^{20} \int_{2}^{\infty} \frac{x^9}{x^{20} - 48x^{10} + 575} dx \right\rfloor $$

Calcular:
$$ \int \frac{dx}{x(x^4 + 1)} $$

Evaluar la siguiente integral indefinida:
$$ \int \frac{dx}{(x - 1)^3(x + 2)^4} $$

Evaluar la integral definida:
$$\int_{1}^{\sqrt{3}} \frac{\arctan x + \text{arccot } x}{x} \, dx$$

Paso 1:
Evaluar: $\int e^{ax+b} \, dx$

Paso 1:
Evaluar: $\int \tan^{-1} \left( \frac{\sin 2x}{1 + \cos 2x} \right) dx$