Results and Formulas of Definite Integrals

1) \int\limits_a^b  {F'(x)dx = F(a) - F(b)} Which is called Fundamental Theorem of Integral Calculus.

2)

\int\limits_a^b  {f(x)dx} = - \int\limits_b^a {f(x)dx}

3)

\int\limits_a^b  {f(x)dx} = \int\limits_a^b {f(t)dt}

4)

\int\limits_a^b  {f(x)dx} = \int\limits_a^c {f(x)dx + }  \int\limits_c^b {f(x)dx} {\text{ , }}a  < c < b

5)

\int\limits_0^a  {f(x)dx} = \int\limits_0^a {f(a - x)dx}

6)

\int\limits_0^{2a}  {f(x)dx} = \int\limits_0^a {f(x)dx} + \int\limits_0^a {f(2a - x)dx}

7) If f(2a - x) = f(x) then

\int\limits_0^{2a}  {f(x)dx} = 2\int\limits_0^a {f(x)dx}

8) If f(2a - x) = - f(x) then

\int\limits_0^{2a}  {f(x)dx} = 0

9) If f(x) = f(a + x) then

\int\limits_0^{na}  {f(x)dx} = n\int\limits_0^a {f(x)dx}

10)

\int\limits_0^{\frac{\pi  }{2}} {\ln (\sin x)dx} =  \int\limits_0^{\frac{\pi }{2}} {\ln (\cos x)dx = - \frac{\pi }{2}\ln 2 = \frac{\pi }{2}\ln  \frac{1}{2}}

11) If f( - x) = f(x) i.e. f(x)is an even function, then

\int\limits_{ - a}^a {f(x)dx}  = 2\int\limits_0^a {f(x)dx}

12) If f( - x) = - f(x) i.e. f(x)is an odd function, then

\int\limits_{ - a}^a {f(x)dx}  = 0

13) If f(x) is a periodic function with period p, i.e.

f(x + p) = f(x)

then for an integer n,

\int\limits_a^{a + n{\text{ }}p} {f(x)dx =  n\int\limits_a^b {f(x)dx} }

14)

\int\limits_0^{\frac{\pi  }{2}} {f(\sin x)dx = } \int\limits_0^{\frac{\pi }{2}} {f(\cos x)dx}

15)

\int\limits_a^b  {f(x)dx = \int\limits_a^b {f(a + b - x)dx} }

16)

\int\limits_a^b  {f(x)dx = \int\limits_0^{b - a} {f(x + a)dx} }

17)

\int\limits_0^{\frac{\pi  }{2}} {\ln \tan xdx = } \int\limits_0^{\frac{\pi }{2}} {\ln \cot xdx = 0}

18)

\int\limits_0^{\frac{\pi  }{2}} {\ln \sec xdx = } \int\limits_0^{\frac{\pi }{2}} {\ln \csc xdx =  \frac{\pi }{2}} \ln 2 = - \frac{\pi  }{2}\ln \frac{1}{2}