If $\int e^{3x} \cos 4x \,dx = e^{3x} (A \sin 4x + B \cos 4x) + c$,then:

Given that $\int_0^\infty \frac{x^2 \, dx}{(x^2 + a^2)(x^2 + b^2)(x^2 + c^2)} = \frac{\pi}{2(a + b)(b + c)(c + a)}$,then the value of $\int_0^\infty \frac{x^2 \, dx}{(x^2 + 4)(x^2 + 9)}$ is

Integrate the function: $\frac{1}{\sqrt{(x-a)(x-b)}}$

If $\int \frac{d \theta}{\cos ^2 \theta(\tan 2 \theta+\sec 2 \theta)}=\lambda \tan \theta+2 \log _{e}|f(\theta)|+c$ (where $c$ is a constant of integration),then the ordered pair $(\lambda, |f(\theta)|)$ is equal to

The value of $I = \int_{0}^{\frac{\pi}{4}} \tan^{n+1} x \, dx + \frac{1}{2} \int_{0}^{\frac{\pi}{2}} \tan^{n-1} \left( \frac{x}{2} \right) \, dx$ is

$\int \frac{2 x+2}{\sqrt{x^2-4 x-5}} d x$ is equal to