Question

how to learn Integration

Answer 1

As many calculus instructors would say, there's no one-size-fits-all approach to solving for integrals. However, that's not to say that a multi-pronged strategy is not possible.

To begin, we can start by building an internal repertoire of integrals which we can easily make sense of:

∫xndx=xn+1n+1∫xndx=xn+1n+1(n≠−1n≠−1)∫1xdx=ln|x|∫1xdx=ln⁡|x|Integrals of the six trigonometric functions (i.e., ∫tanxdx=ln|cosx|∫tan⁡xdx=ln⁡|cos⁡x|)Integrals of exponential functionsand logarithms∫11−x2−−−−−√dx=arcsinx∫11−x2dx=arcsin⁡x∫11+x2=arctanx∫11+x2=arctan⁡x∫1|x|x2−1−−−−−√=arcsec x∫1|x|x2−1=arcsec xIntegrals of hyperbolic functions

Once there, we can extend our integration power by become acquainted with the basic properties of integrals:

∫f±g=∫f±∫g.∫f±g=∫f±∫g.∫kf=k∫f.∫kf=k∫f.∫caf(x)dx=∫baf(x)dx+∫cbf(x)dx.∫acf(x)dx=∫abf(x)dx+∫bcf(x)dx.For an odd function ff, ∫a−af(x)dx=0∫−aaf(x)dx=0.For a function symmetric about cc, ∫c+hc−hf(x)dx=2∫c+hcf(x)dx∫c−hc+hf(x)dx=2∫cc+hf(x)dx

But then, the real meat in integration lies in mastering a series of key techniques of integration:

General Substitution (including the repeated iteration of it, and the version which also substitutes the limits of integration)Reverse Chain-Rule (for functions of the form f(g(x))g′(x)f(g(x))g′(x))Trigonometric Substitution (e.g., sinmxcosnxsinm⁡xcosn⁡x, tanmxsecnxtanm⁡xsecn⁡x, cotmxcscnxcotm⁡xcscn⁡x)Trigonometric Back-Substitution(for functions involving terms like a2−x2−−−−−−√a2−x2, a2+x2a2+x2 and x2−a2x2−a2)Single/Repeated Iterations of Partial Fraction (e.g., products of polynomials, exponential/logarithmic functions and trigonometric functions)Overshooting Method

In many cases, the applications of these techniques can be made easier via the intelligent use of some identities and algebraic manipulation tactics:

ln(ab)=lna+lnbln⁡(ab)=ln⁡a+ln⁡bBasic Trigonometric Identities(i.e., sin2θ+cos2θ=1sin2⁡θ+cos2⁡θ=1, tan2θ+1=sec2θtan2⁡θ+1=sec2⁡θ)Double-Angle FormulasCompleting the SquareQuadratic Factorisation

So, as you can see here, integration draws upon many ancillary skills from both the past and the present. In its core, mastering integration boils down to pattern recognition. That is, recognizing the different functions and the tips/tricks that can be used to integrate them. In fact, the skills and the intuition behind the pattern recognition are mainly developed through repeated exposures and attempts of generalization.

Given its broad nature, it's hence no surprise why many math enthusiasts get excited about solving integrals as a way to sharpen their "mathematical axe". In fact, there are even competitions like Integration Bee that are hosted every single year throughout US - and other countries alike - to promote this culture of problem solving!

Answer 2

HEY THERE ..!!

HERE IS YOURS ANSWER;

★ Integration ★

It is a methematical tool which helps us to get to the Integral or Integrals Value.

◆ This is generally used in Maths & even in some calculation in Physics.


★ How to learn Integration ★

◆ Before learning the Integration, you must be expert in Differentiation. This is backbone of Integration.

◆ Integration is The exactly opposite of Differentiation.


◆ You must be trow with the formulaes of Differentiation as well as integration.

◆ Basically, Calculus is all about Formulae. If the formulaes are well learnt, Then it is like a Cake Walk.



HOPE IT HELPS