If $P = \int_0^{3\pi} f(\cos^2 x) dx$ and $Q = \int_0^{\pi} f(\cos^2 x) dx$,then:

For $0 < a < 1$,the value of the integral $\int_0^\pi \frac{d x}{1-2 a \cos x+a^2}$ is:

The value of $\int_{-\pi / 2}^{\pi / 2} \frac{1}{1+ e^{\sin x}} dx$ is:

If $[t]$ denotes the greatest integer $\leq t$,then the value of $\frac{3(e-1)^2}{e} \int \limits_1^2 x^2 e^{[x]+[x^3]} dx$ is:

Let the domain of the function $f(x) = \log_{4}(\log_{5}(\log_{3}(18x - x^{2} - 77)))$ be $(a, b)$. Then the value of the integral $\int_{a}^{b} \frac{\sin^{3} x}{\sin^{3} x + \sin^{3}(a + b - x)} dx$ is equal to $.....$

Evaluate the definite integral: $\int_{\frac{1}{2}}^{2} \frac{x^2 \ln x}{(1+x^2)^3} dx$