Integration or Integral Calculus is usually passed as the Integral calculus and defined as the reverse operation of the differential calculus. It is true in certain cases but does not capture the true essence of it. Integration is an important part of mathematics that was introduced earlier to differentiation.
\[\large \int_{a}^{\infty}f(x)dx=\lim_{b\rightarrow \infty}\left [ \int_{a}^{b}f(x)dx\right ]\]
\[\large \int_{a}^{b}f(x)dx=F(b)-F(a)\]
a and ∞, b are the lower and upper limits, F(a) is the lower limit value of the integral, F(b) is the upper limit value of the integral.
Definite Integrals Rational or Irrational Expression
\[\large \int_{a}^{\infty }\frac{dx}{x^{2}+a^{2}}=\frac{\pi }{2a}\]
\[\large \int_{a}^{\infty }\frac{x^{m}dx}{x^{n}+a^{n}}=\frac{\pi a^{m-n+1}}{n\sin \left ( \frac{(m+1)\pi }{n} \right )},0< m+1< n\]
\[\large \int_{a}^{\infty }\frac{x^{p-1}dx}{1+x}=\frac{\pi }{\sin (p\pi )},0< p< 1\]
\[\large \int_{a}^{\infty }\frac{x^{m}dx}{1+2x\cos \beta +x^{2}}=\frac{\pi \sin (m\beta )}{\sin (m\pi )\sin \beta }\]
\[\large \int_{a}^{\infty }\frac{dx}{\sqrt{a^{2}-x^{2}}}=\frac{\pi }{2}\]
\[\large \int_{a}^{\infty }\sqrt{a^{2}-x^{2}}dx=\frac{\pi a^{2}}{4}\]
Definite integrals of Trigonometric Functions
\[\large \int_{0}^{\pi }\sin(mx)\sin (nx)dx=\left\{\begin{matrix} 0 & if\;m\neq n\\ \frac{\pi }{2} & if\;m=n \end{matrix}\right.\;m,n\;positive\;integers\]
\[\large \int_{0}^{\pi }\cos (mx)\cos (nx)dx=\left\{\begin{matrix} 0 & if\;m\neq n\\ \frac{\pi }{2} & if\;m=n \end{matrix}\right.\;m,n\;positive\;integers\]
\[\large \int_{0}^{\pi }\sin (mx)\cos (nx)dx=\left\{\begin{matrix} 0 & if\;m+n\;even\\ \frac{2m}{m^{2}-n^{2}} & if\;m+n\;odd \end{matrix}\right.\;m,n\;integers\]
The concept was proposed in 5th century BC where the area of a shape was given by imagining small polygons together. This is a topic of debate usually who discovered Integration actually. Still, the name of Isaac Newton is taken often because he was the person behind the advancements in Integration. In simple words, Integral calculus is the term that is used to calculate the area under a curve.
Moving ahead, Fourier was the person who used the limits to the top or bottom of integral symbol or to mark the start or end point of the integration. This is termed as the definite integral or more applied form of the integration. The concept of definite integrals is frequently used for the real-world problems because it helps to measure or calculate the finite area in a plenty of cases. It is also asked frequently in competitive exams too like JEE or AIEEE etc.
You should be completely focused to solve a problem related to the definite integrals. The only way to succeed in this domain is continuous practice and efforts. Most of the formulas in definite and indefinite integrals are the same but you should know the difference among too when they need to be used. Once you are sure on definite integral properties then solving problems would be easier for you.