Integral of sin x
How to solve the given integral by using the integral of sin x: formula, 1 example, and its solution.
Formula
The derivative of -cos x + C is
-[-sin x] = sin x.
Derivative of cos x
And integral and derivative
are the opposite operations.
The integral of sin x is
-cos x + C.
For an indefinite integral,
there should be +C.
Example∫ (sin x/2 + cos x/2)2 dx
Solution
Solution (Detail)
(sin x/2 + cos x/2)2
= sin2 x/2 + 2(sin x/2)(cos x/2) + cos2 x/2
Square of a Sum
sin2 x/2 + cos2 x/2 = 1
Pythagorean Identity
2(sin x/2)(cos x/2) = sin 2⋅(x/2) = sin x
sin 2A
The integral of 1 is x.
Integral of a Polynomial
The integral of sin x is -cos x.
The given integral is an indefinite integral.
So write +C.
So
x - cos x + C
is the answer.