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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Integration by substitution

Sometimes it is difficult to integrate a function using the simple formulas of integrals. We have to introduce a second variable to make the integration easier.

Example 1

Calculate ∫(x+1)⁵dx

If it was the integral of x⁵ it would be easy to calculate. We can introduce another variable to make the integral easy to calculate. If we make x + 1 equal to u we will just have to calculate the integral of u. So let's write u = x+1

Then our integral becomes ∫u⁵dx. We calculate the integral as if it was the integral of x⁵.

 ∫u⁵dx = u⁵⁺¹/5+1 + C = 1/6 u⁶ + C

Let's substitute u we get  ∫u⁵dx = 1/6 (x+1)⁶ + C

Example 2

Calculate ∫√4x+3dx

Let's write the expression as a power:

 ∫√4x+3dx = ∫(4x+3)¹/² dx

Let's write u = 4x+3 du = 4dx then dx = du/4. Let's substitute 4x+3 and dx

∫√4x+3dx = ∫u¹/².du/4 = 1/4∫u¹/²du = 1/4(u¹/²+¹/¹/²+¹) + C = 1/4(u³/²/3/2) + C = (1/4)(2/3)u³/² + C = 1/6u³/² + C = 1/6√(4x+3)³ + C

If you are interested in learning more about integrals visit Center for Integral Development