$\int_{ - 1}^1 {\log \left( \frac{2 - x}{2 + x} \right)\,dx} = $

Let $T > 0$ be a fixed number. $f: R \rightarrow R$ is a continuous function such that $f(x+T) = f(x)$ for all $x \in R$. If $I = \int_0^T f(x) dx$,then find the value of $\int_0^{5T} f(2x) dx$.

Evaluate the definite integral $\int_{0}^{\frac{\pi}{2}} \frac{\cos ^{2} x}{\cos ^{2} x+4 \sin ^{2} x} d x$.

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:

The value of the definite integral $\int_{2}^{4} (x(3 - x)(4 + x)(6 - x)(10 - x) + \sin x) dx$ equals

Value of the definite integral,$\int\limits_{ - \frac{1}{2}}^{\frac{1}{2}} {\,(\,\,{{\sin }^{ - 1}}(3x - 4{x^3})\,\, - \,\,{{\cos }^{ - 1}}(4{x^3} - 3x)\,\,)\,dx} \,\,$