\displaystyle \textbf{SUMMARY}

\displaystyle \textbf{1.\ } \text{Let }\phi(x)\text{ be the primitive or anti-derivative of a function }f(x)\text{ defined on }[a,b],
\displaystyle \text{i.e., }\frac{d}{dx}\phi(x)=f(x).\text{ Then the definite integral of }f(x)\text{ over }[a,b]\text{ is denoted by }
\displaystyle \int_a^b f(x)\,dx\text{ and is defined as }[\phi(b)-\phi(a)].
\displaystyle \text{i.e., }\int_a^b f(x)\,dx=\phi(b)-\phi(a).
\displaystyle \text{The numbers }a\text{ and }b\text{ are called the limits of integration, }a\text{ is called the lower limit and }b\text{ the}
\displaystyle \text{upper limit. The interval }[a,b]\text{ is called the interval of integration.}

\displaystyle \textbf{2.\ } \text{Following are some fundamental properties of definite integrals which are very} \\ \text{useful in evaluating integrals:}

\displaystyle \textbf{(i)}\ \int_a^b f(x)\,dx=\int_a^b f(t)\,dt,\ \text{i.e., integration is independent of the change of variable.}

\displaystyle \textbf{(ii)}\ \int_a^b f(x)\,dx=-\int_b^a f(x)\,dx,\ \text{i.e., if the limits are interchanged then the value changes} \\ \text{by minus sign only.}

\displaystyle \textbf{(iii)}\ \int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx,\ \text{where }a<c<b.

\displaystyle \int_a^b f(x)\,dx=\int_a^{c_1} f(x)\,dx+\int_{c_1}^{c_2} f(x)\,dx+\cdots+\int_{c_n}^b f(x)\,dx,
\displaystyle \text{where }a<c_1<c_2<\cdots<c_n<b.

\displaystyle \textbf{(iv)}\ \int_0^a f(x)\,dx=\int_0^a f(a-x)\,dx.

\displaystyle \textbf{(v)}\ \int_{-a}^a f(x)\,dx= \begin{cases} 2\int_0^a f(x)\,dx,& \text{if }f(x)\text{ is an even function},\\ 0,& \text{if }f(x)\text{ is an odd function}. \end{cases}

\displaystyle \textbf{(vi)}\ \int_{-a}^a f(x)\,dx=\int_0^a\{f(x)+f(-x)\}\,dx.

\displaystyle \textbf{(vii)}\ \int_0^{2a} f(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}

\displaystyle \textbf{(viii)}\ \int_0^{2a} f(x)\,dx=\int_0^{2a}\{f(x)+f(2a-x)\}\,dx.

\displaystyle \textbf{(ix)}\ \int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx.

\displaystyle \textbf{(x)}\ \int_a^b f(x)\,dx=(b-a)\int_0^1 f\{(b-a)x+a\}\,dx.

\displaystyle \textbf{3.\ } \text{If }f(x)\text{ is a real valued continuous function defined on }[a,b]\text{ which is divided into }n
\displaystyle \text{equal parts each of width }h\text{ by inserting }(n-1)\text{ points }a+h,a+2h,\ldots,a+(n-1)h,
\displaystyle \text{then}
\displaystyle \int_a^b f(x)\,dx=\lim_{h\to0}h\,[\,f(a)+f(a+h)+f(a+2h)+\cdots+f(a+(n-1)h)\,],
\displaystyle \text{where }h=\frac{b-a}{n}.

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