Definite Integration Formulas

Get strong fundamentals of Definite Integration by using the available Definite Integration Formula cheat sheet. With this Definite Integration Formulas list, you can learn definition, properties of definite Integrals, summation of series by intergration, and some other important formulas to solve complicated problems.

1. Definition [NEWTON-LEIBNITZ FORM]

Let ” f” if function of x defined on [a, b] and \(\frac{d}{d x}\) [f(x)] = Φ(x) and a and b, are two values independent of variable x, then for all values of x in domain off then \(\int_{a}^{b}\) Φ(x)dx = \([\mathrm{f}(\mathrm{x})]_{\mathrm{a}}^{\mathrm{b}}\) = f(b) – f(a)

2. Properties of Definite Integral

(i) \(\int_{a}^{b}\)f(x)dx = \(\int_{a}^{b}\)f(t)dt

(ii) \(\int_{a}^{b}\) f(x)dx = – \(\int_{a}^{b}\) f(x)dx

(iii) \(\int_{a}^{b}\)f(x)dx = \(\int_{a}^{c}\)f(x)dx + \(\int_{c}^{b}\)f(x)dx where a < c < b.

(iv) \(\int_{0}^{a}\)f(x)dx = \(\int_{0}^{a}\)f(a – x)dx

(v) \(\int_{-a}^{a}\)f (x) dx = \(\left\{\begin{array}{l}0, \text { if } f(-x)=-f(x) \text { i.e. if } f(x) \text { is odd } \\2 \int_{0}^{a} f(x) d x, \text { if } f(-x)=f(x) \text { i.e.if } f(x) \text { is even }\end{array}\right.\)

(vi) \(\int_{0}^{2 \mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{dx}=\left\{\begin{array}{ll}2 \int_{0}^{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{dx}, & \text { if } \mathrm{f}(2 \mathrm{a}-\mathrm{x})=\mathrm{f}(\mathrm{x}) \\
0, & \text { if } \mathrm{f}(2 \mathrm{a}-\mathrm{x})=-\mathrm{f}(\mathrm{x}) \end{array}\right.\)

(vii) If f(x) = f(x + a), then \(\int_{0}^{na}\)f(x)dx = n\(\int_{0}^{a}\)f(x)dx

(viii) \(\int_{a}^{b}\)f(x)dx = \(\int_{a}^{b}\)f(a + b – x)dx

(ix) \(\frac{d}{d t}\left[\int_{\phi(t)}^{\psi(t)} f(x) d x\right]\) = f{ψ(t)} ψ'(t) – f{Φ(t) Φ'(t)

(x) \(\int_{a+n T}^{b+n T}\)f(x)dx = \(\int_{a}^{b}\)f(x), where f(x) is periodic with period “T”.

(xi) If f(a + n) = f(x) then \(\int_{0}^{na}\) f (x)dx dx = n \(\int_{0}^{a}\) f(x)dx

(xii) \(\int_{ma}^{na}\)f(x) dx = (n – m)\(\int_{0}^{a}\)f(x)dx, where f is periodic function with period “a” i.e. f(a + x) = f(x) ∀ x ∈ [a, b]

(xiii) If f(x) ≤ g(x) then \(\int_{0}^{b}\) f(x)dx ≤ \(\int_{0}^{b}\) g(x)dx

(xiv) \(\left|\int_{a}^{b} f(x) d x\right| \leq \int_{a}^{b}|f(x)| d x\)

(xv) If m is the least value and M is the greastest value of the function f(x) on the interval [a, b] then
m(b – a) ≤ \(\int_{a}^{b}\) f (x)dx ≤ M(b – a)

(xvi) If f(x), g(x) are integrable on the interval (a, b) then
\(\left.\left|\int_{a}^{b} f(x) g(x) d x\right| \leq \sqrt{\left(\int_{a}^{b} f^{2}(x) d x\right)\left(\int_{a}^{b} g^{2}(x) d x\right)}\right)\) above inequality is known as schwarz – Buny ak ovsky inequality.

(xvii) If “f” is continuous, on [a, b] then a Number “c” in [a, b] at which
f(c) = \(\frac{1}{(b-a)} \int_{a}^{b} f(x) d x\),
f(c) is Mean value of function “f in the [a, b]

(xviii) Let p ≤ t ≤ q and a ≤ x ≤ b, also function f(x, t) & f'(x, t) [differentiated with respect to ” t”] are continuous function then I'(t) = \(\int_{a}^{b}\)f'(x, t)dx, were I'(t) is differentiation of I(t) with respect to “t”.

(xix) Improper Integrals:

• \(\int_{a}^{∞}\)f(x)dx = \(\lim _{t \rightarrow+\infty} \int_{a}^{t}\)f(x)dx
• \(\int_{-\infty}^{+a}\) f(x)dx= \(\lim _{t \rightarrow-\infty} \int_{a}^{t}\)f(x)dx
• \(\int_{-\infty}^{+∞}\)f(x)dx= \(\int_{-\infty}^{a}\) f(x)dx + \(\int_{a}^{+\infty}\)f(x)dx
  now use technique of (a) and (b)

3. Some important formulae

I. \(\int_{0}^{\pi / 2}\)logsin x dx = \(\int_{0}^{\pi / 2}\)logcos x dx = –\(\left(\frac{\pi}{2}\right)\)log 2.

II. Walli’s Formula [define of m and n]
(i) \(\int_{0}^{\pi / 2}\)sinⁿx dx = \(\int_{0}^{\pi / 2}\) cosⁿx dx = \(\frac{(n-1)}{n} \frac{(n-3)}{(n-2)} \ldots \frac{2}{3} \cdot 1\)(n is odd)
= \(\frac{(n-1)}{n} \frac{(n-3)}{(n-2)} \ldots . \frac{1}{2} \times \frac{\pi}{2}\)(n is even)

(ii) \(\int_{0}^{\pi / 2}\)sin^mx cosⁿx dx
= \(\frac{(m-1)(m-3) \ldots(2 \text { or } 1)(n-1)(n-3) \ldots(2 \text { or } 1)}{(m+n)(m+n-2) \ldots(2 \text { or } 1)}\) × (1 or π/2)
It is important to note that we multiply by (n/2) when both m and n are even.

4. Summation of series by integration

For finding sum of an infinite series with the help of definite integration, following formula is used
\(\lim _{n \rightarrow \infty} \sum_{r=0}^{n-1} f\left(\frac{r}{n}\right) \cdot \frac{1}{n}=\int_{0}^{1} f(x) d x\)
Working rules

• Express the given series in the form of lim \(\lim _{n \rightarrow \infty} \sum_{r=0}^{n-1} f\left(\frac{r}{n}\right) \cdot \frac{1}{n}\).
• Replace \(\left(\frac{r}{n}\right)\) by x, \(\frac{1}{n}\) by dx, and Σ by ∫, we get the integral ∫ f (x) dx in place of above series.
• The lower limit of this integral = lim \(\lim _{n \rightarrow \infty}\left(\frac{r}{n}\right)_{r=0}\) r is least value in this case r = 0.
• Upper limit = \(\lim _{n \rightarrow \infty}\left(\frac{r}{n}\right)_{r=n-1}\) r is greatest values in this case r = n -1.

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