[V.Rattanapon]

11.\[
\int {\frac{{x^3 + 1}}{{x + 1}}dx = \int {\frac{{\left( {x + 1} \right)\left( {x^2 - x + 1} \right)}}{{x + 1}}} dx} = \int {\left( {x^2 - x + 1} \right)dx = \frac{{x^3 }}{3} - \frac{{x^2 }}{2} + x + c}
\]
12.\[
\int {\frac{{\left( {\sqrt x - 1} \right)^2 }}{{\sqrt x }}dx = \int {\left( {\frac{{x - 2\sqrt x + 1}}{{\sqrt x }}} \right)} } dx = \int {\left( {\sqrt x - 2 + \frac{1}{{\sqrt x }}} \right)dx = \frac{{2x^{\frac{3}{2}} }}{3} - 2x + 2\sqrt x + c}
\]
13.\[
\int {\sec x\left( {\tan x + \cos x} \right)dx = \int {\sec x\tan xdx + \int {\sec x\cos xdx = \sec x + x + c} } }
\]
14.\[
\int {\sqrt {1 - \cos x} dx = \sqrt 2 \int {\sin \frac{x}{2}dx} } = - 2\sqrt 2 \cos \frac{x}{2} + c
\]
15.\[
\int {\left( {e^{2x} + 1} \right)} e^{ - x} dx = \int {\left( {e^x + e^{ - x} } \right)} dx = e^x - e^{ - x} + c
\]
16.\[
\int {\frac{{\tan ax + \tan bx}}{{1 - \tan ax\tan bx}}} dx = \int {\tan \left( {ax + bx} \right)} dx = \frac{1}{{a + b}}\int {\tan \left( {a + b} \right)xd\left( {\left( {a + b} \right)x} \right) = } \frac{1}{{a + b}}\ln \left| {\sec \left( {a + b} \right)x} \right| + c
\]
17.\[
\int {\left( {1 + \cos x} \right)^{\frac{3}{2}} dx = \int {\left( {\sqrt 2 \cos \frac{x}{2}} \right)^3 dx = 2\sqrt 2 \int {\cos ^3 \frac{x}{2}dx = } } } 4\sqrt 2 \int {\left( {1 - \sin ^2 \frac{x}{2}} \right)} d\left( {\sin \frac{x}{2}} \right) = 4\sqrt 2 \left( {\sin \frac{x}{2} - \frac{1}{3}\sin ^3 \frac{x}{2}} \right) + c
\]

18.\[
\int {\frac{{1 + \sin x}}{{1 + \cos x}}} dx = \int {\frac{{\left( {1 + \sin x} \right)\left( {1 - \cos x} \right)}}{{\sin ^2 x}}} dx = \int {\left( {\csc ^2 x - \csc x\cot x + \csc x - \cot x} \right)dx = - \cot x} + \csc x + \ln \left| {\csc x - \cot x} \right| - \ln \left| {\sin x} \right| + c
\]
19.\[
\int {\left( {\sqrt x - \csc ^2 x} \right)} dx = \frac{{2x^{\frac{3}{2}} }}{3} + \cot x + c
\]
20.\[
\int {\frac{1}{{\sqrt x }}} \sec ^2 \sqrt x dx = 2\int {\sec ^2 \sqrt x } d\left( {\sqrt x } \right) = 2\tan \sqrt x + c
\]