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Integration of Trigonometric Functions

Beneath this mask there is more than flesh. Beneath this mask there is an idea, Mr. Creedy, and ideas are bulletproof, V for Vendetta.


In calculus, an antiderivative or indefinite integral, G, of a function g, is the function that can be differentiated to obtain the original function, that is, G’ = f.

G(x) = $\int f(x)dx$

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.


Here are some of the main important trigonometric definitions and formulas:

cos(2θ) = cos2(θ) -sin2(θ) =[sin2(θ) + cos2(θ) = 1] cos2(θ) -(1 - cos2(θ)) = 2cos2(θ) -1 ⇒ cos2(θ) = (1 + cos(2θ))2. Similarly, sin2(θ) = (1 - cos(2θ))2.

sec2(x) = $\frac{1}{cos^2(x)} = \frac{cos^2(x)+sin^2(x)}{cos^2(x)} = 1 + tan^2(x)$. Similarly, $\frac{d}{dx}tan(x) = \frac{d}{dx}\frac{sin(x)}{cos(x)} = \frac{cos^2(x)+sin^2(x)}{cos^2(x)} = \frac{1}{cos^2(x)}=sec^2(x)$. Besides, $\frac{d}{dx} sec(x) = \frac{d}{dx} \frac{1}{cos(x)} = \frac{sin(x)}{cos^2(x)} = sec(x)tan(x).$

Trigonometry substitution for integrals.

Trigonometric substitution is a powerful technique used in calculus to simplify and solve integrals involving radical expressions.

  1. Identify the form of the radical expression in the integral.
  2. Choose a trigonometric function based on the form of the expression, e.g., if $\sqrt{a^2-x^2}$ is present consider using x = acos(θ) or x = asin(θ) to get asin(θ) or acos(θ) respectively. Similarly, if $\sqrt{a^2+x^2}$ is present, consider taking x = atan(θ) to get asec(θ) (Recall sec2(x) = 1 + tan2(x) ⇒ a2+x2 = a2(1 + tan2(θ)) = a2sec2(θ)). Finally, if $\sqrt{x^2-a^2}$ is present consider using x = asec(θ), because x2-a2 = a2sec2(θ) -a2 = a2(sec2(θ)-1) = a2tan2(θ).
  3. Substitute the chosen trigonometric function into the expression, expressing the indefinite variable x in terms of the trigonometric function.
  4. Adjust the differential dx based on the chosen substitution, e.g., if x = asin(θ), dx = acos(θ)dθ.
  5. Substitute and simplify the expression involving trigonometric functions and the adjusted differential into the original integral.
  6. Integrate the simplified expression with respect to the new variable ($\theta$). After integration is completed, back-substitute the original variable x in terms of θ. Simplify the expression further, if necessary, and include any constants of integration.

Completing the square

It is another technique often used in calculus to simplify complex expressions to make integration more manageable.

  1. Identify the “problematic” expression, e.g., if the integral is $\int \frac{dx}{\sqrt{x^2+4x}}$, the difficult expression would be x2 + 4x.
  2. If the coefficient of the quadratic term is not 1, factor it out, that is, divide every term by the coefficient of the quadratic term.
  3. Complete the square, rewrite our expression as (x + a)2 + c, e.g., x2 + 4x = (x + 2)2 -4.
  4. Make the substitution, e.g., u = x +2, du = dx, so x2 + 4x = (x + 2)2 -4 = u2 -4, and rewrite the integral, $\frac{du}{\sqrt{u^2-4}}$.
  5. Simplify the integral, integrate, and back-substitute the original variable. In our case, we use trigonometric substitution, as it was previously mentioned, u = 2sec(θ), du = 2sec(θ)tan(θ)dθ. $\int \frac{du}{\sqrt{u^2-4}}$ =[$u^2-4 = 4sec^2(θ)-4=4(sec^2(θ)-1)=4tan^2(θ)$] $\int \frac{2sec(θ)tan(θ)dθ}{\sqrt{4tan^2(θ)}} = \int \frac{2sec(θ)tan(θ)dθ}{2tan(θ)} = \int sec(θ)dθ$ =[🚀] $ln|sec(θ)+tan(θ)| + C =$[$u=2sec(θ)⇒ sec(θ)=\frac{u}{2}, u^2-4 = 4tan^2(θ)⇒ tan^2(θ) = \frac{u^2-4}{4}$] $ln|\frac{u}{2}+\frac{\sqrt{u^2-4}}{2}| + C$ =[$u = x + 2$] $ln|\frac{x+2}{2}+\frac{\sqrt{x^2+4x}}{2}|+C.$


Now, we need to backtrack our substitution (Figure 3), $x = sin(θ) ⇒ sin(θ)=\frac{x}{1} ⇒ θ = sin^{-1}(x), \int \sqrt{1-x^2}dx = \frac{1}{2}θ + \frac{1}{2}sin(θ)cos(θ)+C = \frac{1}{2}(sin^{-1}(x))+ \frac{1}{2}x\sqrt{1-x^2} + C.$


Now, we need to backtrack our substitution (Figure 2), $\frac{x}{1} = tan(θ).$ By the Pythagorean theorem, $1^2+x^2 = h^2 ⇒ h = \sqrt{1+x^2} ⇒ \int \frac{\sqrt{x^2+1}}{x}dx = ln|csc(θ)-cot(θ)| + sec(θ) + C = ln|\frac{\sqrt{1+x^2}}{x}-\frac{1}{x}|+\frac{\sqrt{1+x^2}}{1}+C = ln|\frac{\sqrt{1+x^2}-1}{x}|+\sqrt{1+x^2}+C.$



This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License and is based on MIT OpenCourseWare [18.01 Single Variable Calculus, Fall 2007].
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