Integrals

Evaluation of Definite Integrals by Substitution


  • When definite integral is to be found by substitution, change the lower and upper limits of integration.
  • If substitution is z = Φ(x) and lower limit of integration is a and upper limit is b, then new lower and upper limits will be Φ(a) and Φ(b) respectively.
  • DEFINITE INTEGRALS - TIPS

    • If a < c < b then \int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx where c ∈ R
    • \int_{a}^{b} f(x) dx = \int_{a}^{c_{1}} f(x) dx + \int_{c_{1}}^{c_{2}} f(x) dx + \int_{c_{2}}^{c_{3}} f(x) dx + ---- + \int_{c_{n}}^{b} f(x) dx
      where a < c1 < c2 < c3 < --- < cn < b
    • If f(x) is a periodic function with period T then \int_{o}^{nT} f(x) dx = n \int_{o}^{T} f(x) dx  ∀ n ∈ N
    • If f(x) is a periodic function with period T and a ∈ R and n ∈ N, then
      \int_{a}^{a + nT} f(x) dx = n \int_{a}^{a + T} f(x) dx = n \int_{o}^{T} f(x) dx
    •  If f(x) is a periodic function with period T and a ∈ R+, then 
      \int_{nT}^{a + nT} f(x) dx = \int_{o}^{a} f(x) dx∀ n ∈ N
    • If f(x) is a periodic function with period T, then \int_{a + T}^{b + T} f(x) dx = \int_{a}^{b} f(x) dx
    • \int_{a + nT}^{b + nT} f(x) dx = \int_{a}^{b} f(x) dx
    • \int_{mT}^{nT} f(x) dx = (n - m) \int_{o}^{T} f(x) dx m, n ∈ Z
    • \frac{d}{dx} \left(\int_{\phi_{1}(x)}^{\phi_{2}(x)} f(t) dt \right)
      = f (\phi_{2} (x)) \phi_{2} \ ^{1} (x) - f (\phi_{1} (x)) \phi_{1} \ ^{1} (x)
    • If f(x) ≤ g(x) on [a, b] then
      \int_{a}^{b} f(x) dx \leq \int_{a}^{b} g(x) dx
    • If m and M are the smallest and greatest values of a function f(x) defined on [a, b] then 
      m (b - a) \leq \int_{a}^{b} f(x) dx \leq M (b - a)
    • \int_{a}^{b} f(x) dx = \int_{a}^{b} f(x) dy = \int_{a}^{b} f(t) dt

View the Topic in this video From 01:00 To 19:40

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Tricks and short cuts on definite integrals

  • \int_{o}^{\pi/2} \sin^{2} x dx = \int_{o}^{\pi/2} \cos^{n}x dx
    \tt = \begin{cases}\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot \frac{n-5}{n-4} ---- \frac{1}{2} \cdot \frac{\pi}{2}; & if \ n \ is \ even \\ \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot \frac{n-5}{n-4} ---- \frac{2}{3} \ \ ; & if \ n \ is \ odd \end{cases}
  • \tt \int_{o}^{\pi/4} \tan^{n} x dx = I_{n} = \frac{1}{n-1} - I_{n-2}
    = \begin{cases}\frac{1}{n-1} - \frac{1}{n-3} +\frac{1}{n-5} - \frac{1}{n-7} + ---- \frac{\pi}{4} ; & if \ n \ is \ even \\ \frac{1}{n-1} - \frac{1}{n-3} + \frac{1}{n-5} - \frac{1}{n-7} ---- + \frac{1}{2} \log 2 \ \ ; & if \ n \ is \ odd \end{cases}
  • I_{n} = \int_{o}^{\pi/4} \sec^{n} x dx = \frac{(\sqrt{2})^{n-2}}{n-1} + \frac{n-2}{n-1} I_{n-2}
  • \int_{o}^{\pi/4} (\tan^{n} x + \tan^{n-2} x) dx = \frac{1}{n-1}
  • \int_{\pi/4}^{\pi/2} (\cot^{n} x + \cot^{n-2} x) dx = \frac{1}{n-1}
  • \int_{o}^{\pi/2} \sin^{m} x \cos^{n}x dx
    \frac{[(m-1) (m-3) (m-5) ----] [(n-1) (n-3) (n-5) ----]}{[(m+n) (m+n-2) (m+n-4) ----]} \cdot \frac{\pi}{2} ; if m and n are even
    \frac{[(m-1) (m-3) ----] [(n-1) (n-3) ----]}{[(m+n) (m+n-2)----]} ; if atleast one of m and n is odd
  •  \int_{o}^{\pi} \sin^{n} x dx = 2 \int_{o}^{\pi} \sin^{n}x dx
  • \int_{o}^{n} [x] dx = \frac{n(n-1)}{2} ∀ n ∈ N
    where [x] is integral part of x
  • \int_{a}^{b} \sqrt{\frac{x-a}{b-x}} \ dx = \frac{\pi}{2} (b - a)
  • \int_{a}^{b} \sqrt{(x-a)(b-x)} dx = \frac{\pi}{8} (b - a)^{2}
  • \int_{a}^{b} \frac{1}{\sqrt{(x-a) (b-x)}} dx = \pi
  • \int_{o}^{\pi/2} \frac{f(\sin x)}{f(\sin x) + f (\cos x)} dx = \frac{\pi}{4}
  • \int_{o}^{\pi/2} \frac{f(\tan x)}{f(\tan x) + f (\cot x)} dx = \frac{\pi}{4}
  • \int_{o}^{\pi/2} \frac{f(\sec x)}{f\left(sec\ x\right)+f(cosec x)} dx = \frac{\pi}{4}
  • \int_{o}^{\pi/2} \frac{a \sin x + b \cos x}{\sin x + \cos x} dx = (a + b) \frac{\pi}{4}
  • \int_{o}^{\pi/2} \frac{a \tan x + b \cot x}{\tan x + \cot x} dx = (a + b) \frac{\pi}{4}
  • \int_{o}^{\pi/2} \log (\sin x) dx = \int_{o}^{\pi/2} \log (\cos x) dx \frac{-\pi}{2} \log 2
  • \int_{o}^{\pi/2} \log (\tan x) dx = \int_{o}^{\pi/2} \log (\cot x) dx = 0
  • \int_{o}^{\pi/2} \frac{dx}{a^{2} \cos^{2} x + b^{2} \sin^{2} x} = \frac{\pi}{2ab}
  • \int_{o}^{\infty} \frac{1}{\left(x + \sqrt{x^{2} + 1}\right)^{n}} dx = \frac{n}{n^{2} - 1}