Question

cot²(3+5x),Integrate the given function defined on proper domain w.r.t. x.

Answer 1

HELLO DEAR,



GIVEN:-
∫cot²(3 + 5x).dx

put (3 + 5x) = t
5.dx = dt
dx = dt/5


therefore, ∫cot²t.dt/5

=> 1/5∫(sec²x - 1).dt

=> 1/5{-cotx - x} + c

=> -1/5(cotx + x) + c.



I HOPE ITS HELP YOU DEAR,
THANKS

Answer 2

Hello,

Answer:
$= - \frac{cot(3 + 5x}{5} - x + c$


Solution:

To solve this integral ,we first convert it into a form which can be integrated.

As we know that cot² x does not have direct formula of integration.

But we know that cosec² x has...

So convert the function into cosec² x, as we know that
$1 + {cot}^{2} x = {cosec}^{2} x \\ \\ {cot}^{2} x = {cosec}^{2} x \: - 1$

Before placing the value ,let

$3 + 5x = t \\ 5dx = dt$
$= \frac{1}{5} {cosec}^{2} t - \frac{1}{5}$


Now,integration of both terms give
$- \frac{1}{5} cot \: t - \frac{t}{5} + c$

undo substitution
$= \frac{ - 1}{5} cot \: (3+ 5x) - \frac{3 + 5x}{5} + c$

is the answer,you can simplify it and send the constant term to C.

=
$- \frac{cot(3 + 5x}{5} - \frac{3}{5} - \frac{5x}{5} + c \\ \\ = - \frac{cot(3 + 5x}{5} - x + c$


is the final answer.

Hope it helps you.