Definite integral

Which of the given values of real numbers $a$, $b\in (0;\frac{\pi }{2})$, such as $a<b$, makes the equality $\underset{a}{\overset{b}{\int }}\cos x\mathrm{d}x=2\cos \frac{\pi }{4}\cdot \sin \frac{\pi }{12}$ true?

The picture shows graphs of two quadratic functions $f_1(x)$ and $f_2(x)$. Find the unknown real positive constant $a$ (as shown in the picture) such that the value of the definite integral $\underset{-1}{\overset{1}{\int }}{f}_1(x)\mathrm{d}x$ is greater by $8$ than the value of the definite integral $\underset{-1}{\overset{1}{\int }}{f}_2(x)\mathrm{d}x$.

Evaluate the definite integral. $$\underset{\mathrm{e}}{\overset{{\mathrm{e}}^2}{\int }}\frac{\ln x}{x}\mathrm{d}x$$

Evaluate the definite integral. $$\underset{-0.5}{\overset{0.1}{\int }}{\mathrm{e}}^{5x-1}\mathrm{d}x$$