integration analysis

Consider the integral

cos(y)³ dy

This is an indefinite integral, so we need to find the most general antiderivative of the integrand f(y).

To find an antiderivative of f(y), we go through our list of integration methods:

Recognize elementary antiderivatives
Rewrite the integrand to make it easier
Use substitution to reverse the chain rule or simplify the integrand
Use integration by parts
Use inspection to see the value of a definite integral

to find one that works for this integrand. In this case, only one method is appropriate. We can rewrite the integrand: split the sine to the third into sine squared (which by basic trigonometry equals one minus cosine squared) and sine to the first to give an expression that is more easily integrable. This is shown below.

Rewriting:
Split the sine to the third into sine squared (which by basic trigonometry equals one minus cosine squared) and sine to the first to rewrite the integral:

cos(y)³ = (1-sin(y)²)cos(y).

Thus,

cos(y)³ dy =

(1-sin(y)²)cos(y)dy

which can be evaluated using substitution, to obtain

(1-sin(y)²)cos(y)dy = (sin(y) - (1/3)sin(y)³) + C.

Explanation for rewritten term(s)
Substitution
Let w = sin(y). Then w' = cos(y), so dw = cos(y) dy. The integral can therefore be rewritten as

(1-sin(y)²)cos(y) dy =

1 - w² dw = w - (1/3)w³ + C

Thus, substituting back for w,

(1-sin(y)²)cos(y) dy = sin(y) - (1/3)sin(y)³ + C

integration tutorial: analysis