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) \[\int\limits_0^{\frac{\pi }{2}} {\ln \sec xdx = } \int\limits_0^{\frac{\pi }{2}} {\ln \csc xdx = \frac{\pi }{2}} \ln 2 = – \frac{\pi }{2}\ln \frac{1}{2}\]