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Index of Definite Integration

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Pattern – 1 : Definite integration never contains any arbitrary constant. i.e.   If  \int f\left(x\right)\ dx = F(x) + c, then \int_{a}^{b}f\left(x\right)\ dx =  F(b) – F(a)

Pattern – 2 : The value of a definite integral does not change with change of variable of integration provided the limits of integration remains the same.

Pattern – 3 : \int_{a}^{b}f\left(x\right)\ dx=\ -\ \int_{b}^{a}f\left(x\right)\ dx

Pattern – 4 : \int_{a}^{b}f\left[g\left(x\right)\right]g^\prime\left(x\right)dx=\ \int_{g\left(a\right)}^{g\left(b\right)}f\left(t\right)dt

Pattern – 5 : If a < c < b, then \int_{a}^{b}f\left(x\right)dx =\ \int_{a}^{c}f\left(x\right)dx+\ \int_{c}^{b}f\left(x\right)dx

Pattern – 6 : \int_{a}^{b}f\left(x\right)\ dx=\ \int_{a}^{b}f\left(a+b-x\right)dx

Pattern – 7 : \int_{a}^{a}f\left(x\right)dx=0

Pattern – 8 : \int_{0}^{b-c}f\left(x+c\right)dx=\ \int_{c}^{b}f\left(x\right)dx

Pattern – 9 : \int_{0}^{a}f\left(x\right)dx =\ \int_{0}^{\frac{a}{2}}f\left(x\right)\ dx+\ \int_{0}^{\frac{a}{2}}f\left(a-x\right)dx

Pattern – 10 : \int_{-a}^{a}f\left(x\right)dx = \begin{cases} & \text{ x } \int_{0}^{a} f(x) dx : \text{f is even} \\ & \text{ 0 : f is odd } \end{cases}

Pattern – 11 : \int_{0}^{2a}f\left(x\right) dx = \int_{0}^{a}[ f(x) + f(2a-x) ] dx deduce that, \int_{0}^{2a}f\left(x\right)dx = \begin{cases} & 2 \int^a_0 f(x) dx : f(2a-x)=f(x)\\ & x= 0 : f(2a-x) = -f(x) \end{cases}

Pattern – 12 : \int_{a}^{b}f\left(x\right)dx = (b-a) \int_{0}^{1}f[ (b-a) x + a ] dx

Pattern – 13 : If f is a periodic function with principal period T, then \int_{0}^{nT}f\left(x\right)\ dx = n \int_{0}^{T}f\left(x\right)\ dx ;   Where, n  I

Pattern – 14 : If f is a periodic function with principal period T, then \int_{mT}^{nT}f\left(x\right)\ dx  =  (n – m) \int_{0}^{T}f\left(x\right)\ dx ;   Where, m, n ∈ I & n > m.

Pattern – 15 : If f is a periodic function with principal period T, then \int_{nT}^{a+nT}f\left(x\right)\ dx  =  \int_{0}^{a}f\left(x\right)\ dx  ;   Where, n ∈ I & a ∈  R

Pattern – 16 : If f is a periodic function with principal period T, then \int_{a+nT}^{b+nT}f\left(x\right)\ dx  = \int_{a}^{b}f\left(x\right)\ dx ;   Where, n ∈ I & a ∈  R

Pattern – 17 : If f is a periodic function with principal period T, then \int_{a}^{a+nT}f\left(x\right)\ dx  =  n \int_{0}^{T}f\left(x\right)\ dx ;   Where, n ∈ I & a ∈  R

Pattern – 18(Even – off function) : If f is a periodic function with principal period T, then\int_{0}^{T}f\left(t\right)dt = \begin{cases} & 2 \int^{T/2}_0 \text{ : f is even } \\ & 0 \text{ : f is odd } \end{cases}

Pattern – 19 ( Odd Function ) : If f is an odd periodic function with principal period T, then    is an even periodic function with principal period T.

Pattern – 20 : If f(x) is an odd periodic defined on interval \left[-\frac{T}{2},\ \ \frac{T}{2}\right] , where T is a principal period of f(x), then  g (x) = \int_{a}^{x}\left(t\right) dt  is periodic function with principal period T. i.e.  g (x + T) = g (x)

Pattern – 21 : If  f(t) is an odd function, then \emptyset\left(x\right)\ =\ \int_{a}^{x}f\ \left(t\right)dt   is an even function.

Pattern – 22 : If f(t) is an even function then g\left(x\right)\ =\ \int_{0}^{x}f\ \left(t\right)dt  is an odd function.

Pattern – 23 : If f(t) is an odd function then g\left(x\right)\ =\ \int_{0}^{x}f\ \left(t\right)dt  is an even function.

Pattern – 24(Leibnitz’s Theorem) : If f is a continuous function on [a, b] and u(x) and v(x) are differentiable functions of x whose values lie in [a, b], then \frac{d}{dx}\left\{\int_{u(x)}^{v(x)}f\left(t\right)\ \ dt\right\} = f{v(x)} \frac{d}{dx} v(x) - f{u(x)} \frac{d}{dx} u(x)

Pattern – 25 (Mean value theorem) : If f is a continuous function on [a, b], then there exists c ∈ (a, b) such that f(c)  =  \frac{1}{b\ -\ a}\int_{a}^{b}f\ \left(t\right)dt

Pattern – 26 : Method to express infinite series as define integrals.

Pattern – 27 : ∀ x ∈ [ a, b ],  g(x) ≤ f(x) ≤ h(x), then \int_{a}^{b}g\left(x\right) dx ≤ \int_{a}^{b}h\left(x\right) dx (where, a<b) deduce that, g(x) ≥ 0, then \int_{a}^{b}g\left(x\right) dx ≥ 0

Pattern – 28 : f is continuous on [a, b], then  \left|\ \int_{a}^{b}f\left(x\right)\ dx\ \right| ≤ \int_{a}^{b}\left|\ f\left(x\right)\ \right|dx

Pattern – 29 : f is continuous on [a, b] and | f(x) | ≤ k, ∀ x ∈ [a, b], then  \left|\int_{a}^{b}f\left(x\right)dx\right| ≤ k (b-a)

Pattern – 30 : If m is the least value (global minimum) and M is greatest value (global maximum) of f(x) for  ∀ x ∈ [a, b] i.e.    m  ≤   f(x)  ≤   M  ;  x  [a, b], then prove that m (b – a) ≤  \int_{a}^{b}f\left(x\right)\ dx   ≤   M (b – a)

Pattern – 31 : If  f^2 (x) and g^2(x) are integrable on the interval [a, b], then prove that \left|\ \int_{a}^{b}f\left(x\right)\ g\left(x\right)\ dx\ \right|    \sqrt{\left(\int_{a}^{b}{f^2\left(x\right)\ dx}\right)\times\left(\int_{a}^{b}{g^2\left(x\right)\ dx}\right)}

Pattern – 32 : I_n=\ \int_{0}^{\pi/4}\tan^n{x} dx then I_n+I_{n-2}=\ \frac{1}{n-1};n\geq 2

Pattern- 33 :  I_n=\ \int_{\pi/4}^{\pi/2}\cot^n{x} dx ,\ then\ I_n+I_{n-2}=\ \frac{1}{n-1};n\geq 2

Pattern – 34 : I_n=\ \int_{0}^{\frac{\pi}{2}}\sin^n{x}dx=\ \int_{0}^{\frac{\pi}{2}}\cos^n{x} = \begin{cases} & \frac{n-1}{n},\frac{n-3}{n-2},\frac{n-5}{n-4},\ldots,\ \frac{1}{2}\times \frac{\pi}{2}\ :n\ is\ even \\ & \frac{n-1}{n}\times\frac{n-3}{n-2}\times\frac{n-5}{n-4}\times\ldots\times\ \frac{2}{3}\times 1\ :n\ is\ odd \end{cases}

Pattern – 35 : I_{m,n}= \int_{0}^{\frac{\pi}{2}}{\sin^mx\ \times \cos^nx\ \ dx} = \begin{cases} & \frac{m-1}{m+n}\times \frac{m-3}{m+n\ -2}\times \frac{m-5}{m+n\ -4}\times ... \times I_{0,n} \text{where m is odd} \\ & \frac{m-1}{m+n}\times \frac{m-3}{m+n\ -2}\times \frac{m-5}{m+n\ -4}\times ... \times I_{1,n} \text{where m is even} \end{cases} Where, I_{0,n} = \int_{0}^{\frac{\pi}{2}}{ \cos^n x\ \ dx}   and  I_{1,n} = \frac{1}{n\ +\ 1}

Pattern – 36 : I_n=\int_{0}^{\infty}{x^n\ e^{-x}\ dx=n!}

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