INTEGRATION

What is Integration?

The process of finding integrals is known as Integration.

Suppose f(x) =dy/dx on the certain interval of the x-axis; then y is called antiderivative or indefinite integral of x and is written as,

\[\displaystyle y=\int{{f(x)dx=\int{{\frac{{dy}}{{dx}}}}}}dx----(1)\]

The given function of the indefinite integral is not unique. That is different from the Integration constant, c.

Definite of f(x) concerning x between the interval, a ≤ x ≤ b

\[\displaystyle \int\limits_{a}^{b}{{f(x)dx=}}h\underset{{}}{\overset{{\lim }}{\longleftrightarrow}}0h\left[ {f(a)+f(a+h)+f(a+2h)+.....+f(a+(n-1)h} \right]--(2)\]

Provided the limit exists.

The quantities a and b are the integral’s lower and upper limits. Symbolically equation (2) is written as,

\[\displaystyle \int\limits_{a}^{b}{{f(x)dx=}}h\underset{{}}{\overset{{\lim }}{\longleftrightarrow}}0h\sum\limits_{{k=0}}^{{h-1}}{{f(a+kh)----(3)}}\]

F(x) is continuous over the interval a ≤ x ≤ b, then definite equation (3) represents the area under the curve y = f(x) bounded by the x-axis and the coordinate at x =a and x=b.

There are some essential theorems for both definite and indefinite integrals;

\[\displaystyle \text{If }g(x)=\int\limits_{a}^{b}{{f(x)dx}},then\frac{{dg(x)}}{{dx}}=f(x)----(4)\]

\[\displaystyle \text{If }f(x)=\frac{{dg(x)}}{{dx}}\text{ is continous in the interval a}\le \text{x}\le \text{b, then,}\]

\[\displaystyle \int\limits_{a}^{b}{{f(x)dx}}=g(x)\int\limits_{a}^{b}{{=g(b)-g(a)----(5)}}\]

Integration Formula

Some integration formulas for the indefinite and the definite integrals, the arbitrary constant of Integration, have been omitted.

(a) Indefinite Integrals

\[\displaystyle \int{{\frac{{df(x)}}{{dx}}dx=f(x)}}\]

\[\displaystyle \int{{\left[ {f(x)\pm g(x)} \right]d(x)=\int{{f(x)dx}}}}\pm \int{{g(x)dx}}\]

\[\displaystyle \int{{cf(x)dx=c}}\int{{f(x)dx}}\]

\[\displaystyle \int{{{{x}^{n}}dx=\frac{{{{x}^{{n+1}}}}}{{n+1}};n\ne -1}}\]

\[\displaystyle \int{{\sin xdx=-\cos x}}\]

\[\displaystyle \int{{\cos xdx=\sin x}}\]

(b) Definite integrals

\[\displaystyle \int\limits_{0}^{\infty }{{{{e}^{{-a{{x}^{2}}}}}}}dx=\frac{1}{2}{{\left( {\frac{\pi }{a}} \right)}^{{\frac{1}{2}}}}\]

\[\displaystyle \int\limits_{0}^{\infty }{{{{x}^{n}}}}{{e}^{{-ax}}}dx=\frac{{n!}}{{{{a}^{{n+1}}}}}\]

\[\displaystyle \int\limits_{0}^{\infty }{{{{x}^{n}}}}{{e}^{{-a{{x}^{2}}}}}dx=\frac{{(1)(3)(5).....(2n-1)}}{{{{2}^{{n+1}}}{{a}^{n}}}}{{\left( {\frac{\pi }{a}} \right)}^{{\frac{1}{2}}}}\]

\[\displaystyle \int\limits_{0}^{\infty }{{{{x}^{n}}}}{{e}^{{-a{{x}^{2}}}}}dx=\frac{1}{{2a}}\]

\[\displaystyle \int\limits_{0}^{a}{{f(x)dx=\int\limits_{0}^{a}{{f(x-a)dx}}}}\]

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