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,

• Integration Formulas
• Integration by Substitution
• Integration by Parts

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.