How to Apply Trigonometric Substitution Method?
We can apply the trigonometric substitution method as discussed below,
Integral with a2 – x2
Let’s consider an example of the Integral involving a2 – x2.
Example: [Tex]\int \frac{1}{\sqrt{a^2-x^2}}\hspace{0.1cm}dx[/Tex]
Lets put, x = a sinθ
⇒ dx = a cosθ dθ
Thus, I = [Tex]\int \frac{a\hspace{0.1cm}cos\theta \hspace{0.1cm}d\theta}{\sqrt{(a^2-(a\hspace{0.1cm}sin\theta)^2)}}[/Tex]
⇒ I = [Tex]\int \frac{a\hspace{0.1cm}cos\theta \hspace{0.1cm}d\theta}{\sqrt{(a^2cos^2\theta)}}[/Tex]
⇒ I = [Tex]\int 1. d\theta[/Tex]
⇒ I = θ + c
As, x = a sinθ
⇒ θ = [Tex]sin^{-1}(\frac{x}{a})[/Tex]
⇒ I = [Tex]sin^{-1}(\frac{x}{a}) + c[/Tex]
Integral with x2 + a2
Let’s consider an example of the Integral involving x2 + a2.
Example: Find the integral [Tex]\bold{\int \frac{1}{x^2+a^2}\hspace{0.1cm}dx}[/Tex]
Solution:
Lets put x = a tanθ
⇒ dx = a sec2θ dθ, we get
Thus, I = [Tex]\int \frac{1}{(a\hspace{0.1cm}tan\theta)^2+a^2}\hspace{0.1cm}(a\hspace{0.1cm}sec^2\theta \hspace{0.1cm}d\theta)[/Tex]
⇒ I = [Tex]\int \frac{a\hspace{0.1cm}sec^2\theta \hspace{0.1cm}d\theta}{a^2(sec^2\theta)}[/Tex]
⇒ I = [Tex]\frac{1}{a}\int 1.d\theta[/Tex]
⇒ I = [Tex]\frac{1}{a} \theta [/Tex]+ c
As, x = a tanθ
⇒ θ = [Tex]tan^{-1}(\frac{x}{a})[/Tex]
⇒ I = [Tex]\frac{1}{a}tan^{-1}(\frac{x}{a}) [/Tex]+ c
Integral with a2 + x2.
Let’s consider an example of the Integral involving a2+ x2.
Example: Find the integral of [Tex]\bold{\int \frac{1}{\sqrt{a^2+x^2}}\hspace{0.1cm}dx}[/Tex]
Solution:
Lets put, x = a tanθ
⇒ dx = a sec2θ dθ
Thus, I = [Tex]\int \frac{a\hspace{0.1cm}sec^2\theta \hspace{0.1cm}d\theta}{\sqrt{(a^2+(a\hspace{0.1cm}tan\theta)^2)}}[/Tex]
⇒ I = [Tex]\int \frac{a\hspace{0.1cm}sec^2\theta \hspace{0.1cm}d\theta}{\sqrt{(a^2\hspace{0.1cm}sec^2\theta)}}[/Tex]
⇒ I = [Tex]\int \frac{a\hspace{0.1cm}sec^2\theta \hspace{0.1cm}d\theta}{a\hspace{0.1cm}sec\theta}[/Tex]
⇒ I = [Tex]\int sec\hspace{0.1cm}\theta d\theta[/Tex]
⇒ I = [Tex]log|sec\hspace{0.1cm}\theta+tan\hspace{0.1cm}\theta| + c[/Tex]
⇒ I = [Tex]log|tan\hspace{0.1cm}\theta+\sqrt{1+tan^2\hspace{0.1cm}\theta}| + c[/Tex]
⇒ I = [Tex]log|\frac{x}{a}+\sqrt{1+\frac{x^2}{a^2}}|+ c[/Tex]
⇒ I = [Tex]log|\frac{x}{a}+\sqrt{\frac{a^2+x^2}{a^2}}|+ c[/Tex]
⇒ I = [Tex]log|\frac{x}{a}+\frac{1}{{a}}\sqrt{a^2+x^2}|+ c[/Tex]
⇒ I = [Tex]log|x+\sqrt{a^2+x^2}|-log\hspace{0.1cm}a+ c[/Tex]
⇒ I = [Tex]log|x+\sqrt{a^2+x^2}|+ c_1[/Tex]
Integral with x2 – a2.
Let’s consider an example of the Integral involving x2 – a2.
Example: Find the integral of [Tex]\bold{\int \frac{1}{\sqrt{x^2-a^2}}\hspace{0.1cm}dx}[/Tex]
Let’s put, x = a secθ
⇒ dx = a secθ tanθ dθ
Thus, I = [Tex]\int \frac{a\hspace{0.1cm}sec\theta \hspace{0.1cm}tan\theta\hspace{0.1cm}d\theta}{\sqrt{((a\hspace{0.1cm}sec\theta)^2-a^2)}}[/Tex]
⇒ I = [Tex]\int \frac{a\hspace{0.1cm}sec\theta \hspace{0.1cm}tan\theta\hspace{0.1cm}d\theta}{(a\hspace{0.1cm}tan\theta)}[/Tex]
⇒ I = [Tex]\int sec\theta\hspace{0.1cm}d\theta[/Tex]
⇒ I = [Tex]log|sec\hspace{0.1cm}\theta+tan\hspace{0.1cm}\theta| + c[/Tex]
⇒ I = [Tex]log|sec\hspace{0.1cm}\theta+\sqrt{sec^2\hspace{0.1cm}\theta-1}| + c[/Tex]
⇒ I = [Tex]log|\frac{x}{a}+\sqrt{\frac{x^2}{a^2}-1}|+ c[/Tex]
⇒ I = [Tex]log|\frac{x}{a}+\sqrt{\frac{x^2-a^2}{a^2}}|+ c[/Tex]
⇒ I = [Tex]log|\frac{x}{a}+\frac{1}{{a}}\sqrt{x^2-a^2}|+ c[/Tex]
⇒ I =[Tex] log|x+\sqrt{x^2-a^2}|-log\hspace{0.1cm}a+ c[/Tex]
⇒ I = [Tex]log|x+\sqrt{x^2-a^2}|+ c_1[/Tex]
Read More,
Trigonometric Substitution: Method, Formula and Solved Examples
Trigonometric Substitution is one of the substitution methods of integration where a function or expression in the given integral is substituted with trigonometric functions such as sin, cos, tan, etc. Integration by substitution is an easiest substitution method.
It is used when we make a substitution of a function, whose derivative is already included in the given integral function. By this, the function gets simplified, and simple integrals function is obtained which we can integrate easily. It is also known as u-substitution or the reverse chain rule. Or in other words, using this method, we can easily evaluate integrals and antiderivatives.