Question

Are there any method apart from integration by parts , u substituting, trig substitution or partial decomposition for difficult integrals?

Answer 1

See explanation

Explanation:

The methods mentioned above are the most powerful/useful, and many integrals can be solved using these methods, sometimes multiple times, and with the right algebraically manipulations

However, here are some thoughts on some other methods (for undefinite integrals), but unfortunately their are no magic formula, when we dealing with integrals. (Which makes it fun and challenging)

Closely related to trig substitution, hyperbolic substitution,
which follow the same principles as trig substitution, just with hyperbolic functions instead of trig functions

Another powerful substitution method related to, trig substitution,
is tangent half angle substitution,
This turns a trig integral, into an integral involving polynomials. Which often is easier (however they can be quite difficult)
Making the substitution `u=tan(x/2)`, the expressions from the scheme can be derived

Let's take an example

`I=int1/(1+sin(x))dx`

`color(white)(I)=int1/(1+(2u)/(u^2+1))2/(u^2+1)du`

`color(white)(I)=2int1/(u+1)^2du`

`color(white)(I)=-2/(tan(x/2)+1)+C`

Many integrals can't be evaluated using elementary functions. Some of the most basic have been given names, such as the gamma function `Gamma(z)` and the sine integral `"Si"(z)`

Expressing the integral as a series is sometimes useful here, when the integral can't be expressed using elementary functions

Let's take an example

`I=intsin(x)/x`

`color(white)(I)=int(x-x^3/(3!)+x^5/(5!)+...)/xdx`

`color(white)(I)=int1-x^2/(3!)+x^4/(5!)+...dx`

`color(white)(I)=(x-x^3/(3*3!)+x^5/(5*5!)+...)+C`

`color(white)(I)=sum_(n=0)^oo((-1)^nx^(2n+1))/((2n+1)(2n+1)!)+C`

There are some other worthy mentions, for example the use complex numbers and Euler's formula is often useful for some integrals

Here are some more on the topic here and here

For definite integrals there are even more techniches, for example differentiation under the integral sign

Answer 2

In addition there are many techniques for evaluating definite integrals even though we cannot find an elementary anti derivative for the integrand. Such techniques could use Complex Variable Theory, Asymptotics or change of coordinate system.

A classic example is evaluation of the Gaussian Integral :

`int_(-oo)^(oo) \ e^(x^2) \ dx = sqrt(pi)`