Question

If the value of the integral ${\int }_0^{\frac{\pi }{2}}cot\frac{\theta }{2}\left(1-{\left(cos\right)}^4\theta \right)d\theta$ is equal to $\lambda ,$ then the value of $3\lambda$ is equal to

Answer 1

Answer: 6.25

Let $I={\int }_0^{\frac{\pi }{2}}cot\frac{\theta }{2}\left(1-cos\theta \right)\left(1+cos\theta +co{s}^2\theta +co{s}^3\theta \right)d\theta$
So, $I={\int }_0^{\frac{\pi }{2}}\frac{cos\frac{\theta }{2}}{sin\frac{\theta }{2}}\cdot 2si{n}^2\frac{\theta }{2}\left(1+cos\theta +{\left(cos\right)}^2\theta +{\left(cos\right)}^3\theta \right)d\theta$
$={\int }_0^{\frac{\pi }{2}}sin\theta \left(1+cos\theta +{\left(cos\right)}^2\theta +{\left(cos\right)}^3\theta \right)d\theta$
Let $cos\theta =t$
$\Rightarrow I={\int }_0^1\left(1+t+t^2+t^3\right)dt=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}=\frac{25}{12}$
Hence, $\lambda =\frac{25}{12}\Rightarrow 3\lambda =\frac{25}{4}=6.25$