Chapter – Indefinite Integral
Sub Topic – Integration by Parts
Outline
Integration of rational function involving sinx and cosx of the type
INTEGRAL OF FUNCTIONS WHICH ARE RATIONAL IN Sin x and Cos x
Problem 1:
Solution: put tan
(dx=
Cos x =
INTEGRATION BY PARTS
If u and v are any two differentiable variable of a single variable x (say).
Then, by the product rule of differentiation, we have
+ v
Integrating both sides, we get
uv =
or
...(1)
u = f(x) and
Let
Therefore, expression (1) can be rewritten as
i.e.,
The integral of the product of two functions =
(first function) x (integral of the function)
– Integral of [(differential coefficient of the first function) x
(integral of the second function)]
PROBLEM Find
Solution: Put f (x) = x (first function) and g (x) = cos x (second function).
Then integration by parts gives
= x sin x –
Problem :
Solution: u = x, dv = sin 3x dx
=
Problem :
Solution: let u =
=
Problem :
Solution: Take u = log x, and v = x => dv = 1 dx
.1 dx
= x log x – x + C
= x (log x – 1) + C