Formulas for Definite Integrals | Newton Leibnitz Formula

Here, you will learn formulas for definite integrals and properties of definite integrals and newton leibnitz formula with examples.

Let’s begin –

A definite integral is denoted by \(\int_{a}^{b}\) f(x)dx which represent the algebraic area bounded by the curve y = f(x), the ordinates x = a, x = b and the x-axis.

Properties and Formulas for Definite Integrals

(a)  \(\int_{a}^{b}\) f(x)dx = \(\int_{a}^{b}\) f(t)dt provided f is same

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

(c)  \(\int_{a}^{b}\) f(x)dx = \(\int_{a}^{c}\) f(x)dx + \(\int_{c}^{b}\) f(x)dx , where c may lie inside or outside the interval [a,b]. This property is to be used when f is piecewise continous in (a, b).

(d)  \(\int_{a}^{a}\) f(x)dx = \(\int_{0}^{a}\) [f(x) + f(-x)]dx = \(\begin{cases} 0 & \text{if f(x) is an odd function}\ \\ 2\int_{a}^{b} f(x)dx & \text{if f(x) is an even function}\ \end{cases}\)

Example : Evaluate \(\int_{1/2}^{1/2}\) \(cosx ln{({1+x\over 1-x})}\) dx

Solution : f(-x) = \(cos(-x) ln{({1-x\over 1+x})}\) = – \(cosx ln{({1+x\over 1-x})}\) = f(-x)

\(\implies\)   f(x) is odd

Hence, the value of the given interval is 0.

(e)  \(\int_{a}^{b}\) f(x)dx = \(\int_{a}^{b}\) f(a+b-x)dx, In particular \(\int_{0}^{a}\) f(x)dx = \(\int_{0}^{a}\) f(a-x)dx

Example : Evaluate \(\int_{0}^{\pi/2}\) \(asinx+bcosx\over sinx+cosx\) dx

Solution : I = \(\int_{0}^{\pi/2}\) \(asinx+bcosx\over sinx+cosx\) dx     ….(i)

I = \(\int_{0}^{\pi/2}\) \(asin(\pi/2-x)+bcos(\pi/2-x)\over sin(\pi/2-x)+cos(\pi/2-x)\) dx = \(\int_{0}^{\pi/2}\) \(acosx+bsinx\over sinx+cosx\) dx     ….(ii)

Adding (i) and (ii),

2I = \(\int_{0}^{\pi/2}\) \(a+b)(sinx+cosx)\over sinx+cosx\) dx = \(\int_{0}^{\pi/2}\) (a+b) dx = (a+b)\(\pi/2\)

\(\implies\)   I = (a+b)\(\pi/4\)

(f)  \(\int_{0}^{2a}\) f(x)dx = \(\int_{0}^{a}\) f(x)dx + \(\int_{0}^{a}\) f(2a-x)dx = \(\begin{cases} 2\int_{0}^{a} f(x)dx & \text{if}\ f(2a-x) = f(x) \\ 0 & \text{if}\ f(2a-x) = -f(x) \end{cases}\).

(g)  \(\int_{0}^{nT}\) f(x)dx = n\(\int_{0}^{T}\) f(x)dx, (n \(\in\) I); where T is the period of the function i.e. f(T+x) = f(x)

(h)  \(\int_{a+nT}^{b+nT}\) f(x)dx = \(\int_{a}^{b}\) f(x)dx, where f(x) is periodic with period T & n \(\in\) I.

(i)  \(\int_{mT}^{nT}\) f(x)dx = (n-m)\(\int_{0}^{T}\) f(x)dx, where f(x) is periodic with period T & (n, m \(\in\) I).

Walli’s formula

If m,n \(\in\) N & m, n \(\ge\) 2, then

(a)  \(\int_{0}^{\pi/2}\) \(sin^nx\)dx = \(\int_{0}^{\pi/2}\) \(cos^nx\)dx = \((n-1)(n-3)….(1 or 2)\over {n(n-2)….(1 or 2)}\) K 

where K = \(\begin{cases} \pi/2 & \text{if n is even}\ \\ 1 & \text{if n is odd}\ \end{cases}\)

(b)  \(sin^nx.cos^mx\)dx = \([(n-1)(n-3)….(1 or 2)][(m-1)(m-3)….(1 or 2)]\over {(m+n)(m+n-2)(m+n-4)….(1 or 2)}\) K

where K = \(\begin{cases} \pi/2 & \text{if both m and n are even}\ \\ 1 & \text{otherwise}\ \end{cases}\).

Newton-Leibnitz formula

If h(x) and g(x) are differentiable functions of x then,

\(d\over dx\) \(\int_{g(x)}^{h(x)}\) f(t)dt = f[h(x)].h'(x) – f[g(x)].g'(x)

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