Question

Methods in which we reduce the function standard forms for finding the integrals are:
a) Integration by parts
b) Integration using partial fraction
c) Integration by substitution
d) All the above

Answer 1

Hint: Check every method and observe if the given method is capable of reducing the integral function into standard form so that it can be solved. If after altering the integral function using respective methods can be solved then it must be the method of reduction of function into standard form.

Complete step-by-step answer:
Integration by part, integration using partial fraction, integration by substitution are used to reduce the function into integrals as:
In integration by part,
$\int {uvdx\; = u\int {vdx\; - \int {u\prime (\int v dx)dx} } } $
We reduce the two-function consisting equation into standard form.

Integration using partial fraction,
It’s used when fraction is in improper form and we can convert into proper fraction using this method.

Integration by substitution,
In this method we use when the integrand consists of two functions in the expression and one function is derivative of the other.
$\int {uv\;dx} $, such that $\dfrac{{du}}{{dx}} = v$
Hence, the correct option is (d).

Note: We have to check which operation reduces the expression in standard form. We should know how to use the operations to find the same like for Integration by parts solving, ILATE method gives an idea for choosing functions.