જો $I = \int\limits_0^{\frac{\pi}{2}} \ln(\sin x) dx$ હોય,તો $\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \ln(\sin x + \cos x) dx =$

$\left[ {\sum\limits_{n = 1}^{10} {\int_{ - 2n - 1}^{2n} {{{\sin }^{27}}x\,dx} } } \right] + \left[ {\sum\limits_{n = 1}^{10} {\int_{2n}^{2n + 1} {{{\sin }^{27}}x\,dx} } } \right]$ ની કિંમત શોધો.

$\int_{0}^{\pi / 2} \log (\operatorname{cosec} x) d x$ નું મૂલ્ય શોધો.

ધારો કે $I=\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{1}{2-\cos 2 x}\left(\frac{3}{\pi}+\log \left(\frac{4+\sin x}{4-\sin x}\right)\right) d x$. આપેલ છે કે $\int \frac{d x}{1+k x^2}=\frac{1}{\sqrt{k}} \tan ^{-1}(\sqrt{k} x)+c, \tan ^{-1}(0)=0$ અને $\tan ^{-1}(\sqrt{3})=\frac{\pi}{3}$. તો $3 I^2=$

જો $P = \int_0^{3\pi} f(\cos^2 x) dx$ અને $Q = \int_0^{\pi} f(\cos^2 x) dx$ હોય,તો:

$\int_{\frac{-3\pi}{2}}^{\frac{-\pi}{2}} \left[ (x+\pi)^3 + \cos^2(x+3\pi) \right] dx = $