Prove via differentiation that integral $$\int_0^\pi \frac{\log(1+\cos\alpha\cos\theta)}{\cos\theta}\,d\theta$$ is equal to $$ \pi\left(\frac{\pi}{2} - \alpha\right) $$ where $0\leq \alpha\leq \frac{\pi}{2}.$
After partially differentiating wrt $\alpha$ I have tried via substitution;
$$u=\log(1+\cos\alpha\cos\theta)$$
Then I get;
$$\frac{dI(\alpha)}{d\alpha}=-\sin\alpha\int_0^\pi \frac{1}{1+\cos\alpha\cos\theta}\,d\theta$$
I can not currently see how to progress this to stated answer.
Please show full working.
The question is from G Stephenson's 'Mathematical Methods for Science students' - page 172.