Application of the integration by substitution to trigonometry integrals

The integration by substitution method can be used to find the integral of some trigonometric expressions. Here are the basic trigonometric formulas that can be used with the substitution method.
The change of variable u can be applied as long as it is a differentiable function of x

Example I

Solution

The argument of the cosine function is 3x + 2, Let's make u = 3x + 2. then du = 3 dx  dx = 1/3du
Let;s substitute dx in the given expression:

                ∫ cos(3x + 2) dx = ∫cosu.1/3du

                                          = 1/3∫ cosudu

                                          = 1/3sinu + C

                                            = 1/3sin (3x +2) + C

Example II

Solution

Example III

Solution

Let's substitute tanx by sinx/cosx in the expression:

∫tanxdx = ∫ cosx/sinxdx

Let's u = sinx. Then du = cosxdx or dx = du/cosx

Let's substitute u and dx in the given expression

∫cosx/sinxdx = ∫cosx/u.du/cosx

Let's simplify by cosx:

∫cosx/sinxdx = ∫du/u = lnu + C = ln sinx + C

Practice

Evaluate:

1) ∫sin(4x + 1)dx

2) ∫1/sin²xdx

3) ∫cotx

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