What is the value of $\int_{0}^{10} \frac{x+2}{x+1}\, dx$? [3 points]
(1) $10 + \ln 5$
(2) $10 + \ln 7$
(3) $10 + 2\ln 3$
(4) $10 + \ln 11$
(5) $10 + \ln 13$

We define $E _ { 1 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p = q \right\}$ and $S _ { p , q } = \int _ { 0 } ^ { 1 } \dfrac { t ^ { q - 1 } } { 1 + t ^ { p } } d t$.
Show that, for all $( p , q ) \in E _ { 1 }$, $$S _ { p , q } = \frac { \ln 2 } { p }$$

We define $E _ { 2 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p < q , p \mid q \right\}$ and $S _ { p , q } = \int _ { 0 } ^ { 1 } \dfrac { t ^ { q - 1 } } { 1 + t ^ { p } } d t$.
For all pairs $( p , q ) \in E _ { 2 }$, show that there exists a constant $\lambda := \lambda ( p , q )$ which one will determine, such that $$S _ { p , q } = \frac { ( - 1 ) ^ { \lambda - 1 } } { p } \left( \ln ( 2 ) - \sum _ { k = 1 } ^ { \lambda - 1 } \frac { ( - 1 ) ^ { k - 1 } } { k } \right)$$

$\int \frac { \sin ^ { 8 } x - \cos ^ { 8 } x } { \left( 1 - 2 \sin ^ { 2 } x \cos ^ { 2 } x \right) } d x$ is equal to
(1) $- \frac { 1 } { 2 } \sin 2 x + c$
(2) $- \sin ^ { 2 } x + c$
(3) $- \frac { 1 } { 2 } \sin x + c$
(4) $\frac { 1 } { 2 } \sin 2 x + c$

The integral $\int \sqrt { 1 + 2 \cot x ( \operatorname { cosec } x + \cot x ) } d x , \left( 0 < x < \frac { \pi } { 2 } \right)$ is equal to
(1) $2 \log \left| \sin \frac { x } { 2 } \right| + c$
(2) $4 \log \left| \sin \frac { x } { 2 } \right| + c$
(3) $4 \log \left| \cos \frac { x } { 2 } \right| + c$
(4) $2 \log \left| \cos \frac { x } { 2 } \right| + c$

If $f ( t ) = \int _ { 0 } ^ { \pi } \frac { 2 x \mathrm {~d} x } { 1 - \cos ^ { 2 } \mathrm { t } \sin ^ { 2 } x } , 0 < \mathrm { t } < \pi$, then the value of $\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \pi ^ { 2 } \mathrm { dt } } { f ( \mathrm { t } ) }$ equals $\_\_\_\_$

$$\int_{0}^{\frac{\pi}{3}} \frac{\sin x}{\cos^{2} x}\, dx$$
What is the value of the integral?
A) 2
B) 1
C) 0
D) $-1$
E) $-2$

$$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin 2 x \cdot \cot x \, d x$$
What is the value of this integral?
A) $\frac { \pi + 1 } { 2 }$
B) $\frac { \pi + 1 } { 3 }$
C) $\frac { \pi + 2 } { 4 }$
D) $\frac { \pi - 1 } { 6 }$
E) $\frac { \pi - 2 } { 6 }$