14.0 Important Results

1. $\int\limits_0^{{\pi \over 2}} {{{{{\sin }^n}x} \over {{{\sin }^n}x + {{\cos }^n}x}}dx} = \int\limits_0^{{\pi \over 2}} {{{{{\cos }^n}x} \over {{{\sin }^n}x + {{\cos }^n}x}}dx} = {\pi \over 4}$

2. $\int\limits_0^{{\pi \over 2}} {{{{{\tan }^n}x} \over {{{\tan }^n}x + {{\cot }^n}x}}dx} = \int\limits_0^{{\pi \over 2}} {{{{{\cot }^n}x} \over {{{\tan }^n}x + {{\cot }^n}x}}dx} = {\pi \over 4}$

3. $\int\limits_0^{{\pi \over 2}} {\log \sin xdx} = \int\limits_0^{{\pi \over 2}} {{\mathop{\rm logcos}\nolimits} xdx} = - {\pi \over 2}\log 2$

4. $\int\limits_0^{{\pi \over 2}} {{\mathop{\rm logtan}\nolimits} xdx} = \int\limits_0^{{\pi \over 2}} {{\mathop{\rm logcot}\nolimits} xdx} = 0$

5. $\int\limits_0^{{\pi \over 2}} {\log \sec xdx} = \int\limits_0^{{\pi \over 2}} {{\mathop{\rm logcosec}\nolimits} xdx} = {\pi \over 2}\log 2$

6. $\int\limits_0^\infty {{e^{ - ax}}\sin bxdx} = {b \over {{a^2} + {b^2}}}$

7. $\int\limits_0^\infty {{e^{ - ax}}\cos bxdx} = {a \over {{a^2} + {b^2}}}$