Question

sin²(3x+5),Integrate the given function defined on proper domain w.r.t. x.

Answer 1

In the attachment I have answered this problem.                         I have applied decomposition method to find the anti derivative of the given function.                     The given  non-integrable function is changed into integrable function by using trigonometric formula.                        See the attachment for detailed solution.

Answer 2

Hello,

Answer:
$integration \: \: \: of \: {sin}^{2} (3x + 5) \: \: wrt \: \: x \: \\ = \: \frac{x}{2} - \frac{1}{12} sin(6x + 10) + c$

Solution:

1) first convert the function into integrate able form

As we know that

$\frac{1 - \cos(2x) }{2} = { sin}^{2} x$
So,apply this to the given function

To integrate

$\frac{1 - cos(2(3x + 5))}{2} dx\\ \\ = \frac{1}{2} dx \: \: - \frac{1}{2} cos(6x + 10) \: dx$

Now integrate the function

we get

$\frac{x}{2} - \frac{1}{12} sin(6x + 10) + c$

as integration of constant term is 1, and cos (6x+10) is sin(6x+10) and coefficient of x in denominator.

Hope it helps you.