# Results and Formulas of Definite Integrals

1) $\int\limits_a^b {F'(x)dx = F(a) – F(b)}$  is called the 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) $\int\limits_0^{\frac{\pi }{2}} {\ln \sec xdx = } \int\limits_0^{\frac{\pi }{2}} {\ln \csc xdx = \frac{\pi }{2}} \ln 2 = – \frac{\pi }{2}\ln \frac{1}{2}$