Fundamental Formulae of Indefinite Integration (Inverse Trigonometric Functions) - Practice Questions & MCQ

Fundamental Formulae (Inverse Trigonometric Functions)

1. $\frac{d}{d x}\left(\sin ^{-1} \frac{x}{a}\right)=\frac{1}{\sqrt{a^2-x^2}} \Rightarrow \int \frac{d x}{\sqrt{a^2-x^2}}=\sin ^{-1}\left(\frac{x}{a}\right)+C$
2. $\frac{\mathrm{d}}{\mathrm{dx}}\left(\cos ^{-1} \frac{\mathrm{x}}{\mathrm{a}}\right)=\frac{-1}{\sqrt{\mathrm{a}^2-\mathrm{x}^2}} \Rightarrow \int \frac{-1}{\sqrt{\mathrm{a}^2-\mathrm{x}^2}} \mathrm{dx}=\cos ^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right)+\mathrm{C}$
3. $\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{1}{\mathrm{a}} \tan ^{-1} \frac{\mathrm{x}}{\mathrm{a}}\right)=\frac{1}{\mathrm{a}^2+\mathrm{x}^2} \Rightarrow \int \frac{\mathrm{dx}}{\mathrm{a}^2+\mathrm{x}^2}=\frac{1}{\mathrm{a}} \tan ^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right)+\mathrm{C}$
4. $\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{1}{\mathrm{a}} \cot ^{-1} \frac{\mathrm{x}}{\mathrm{a}}\right)=\frac{-1}{\mathrm{a}^2+\mathrm{x}^2} \Rightarrow \int \frac{-1}{\mathrm{a}^2+\mathrm{x}^2} \mathrm{dx}=\frac{1}{\mathrm{a}} \cot ^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right)+\mathrm{C}$
5. $\frac{d}{d x}\left(\frac{1}{a} \sec ^{-1} \frac{x}{a}\right)=\frac{1}{x \sqrt{x^2-a^2}} \Rightarrow \int \frac{d x}{x \sqrt{x^2-a^2}}=\frac{1}{a} \sec ^{-1}\left(\frac{x}{a}\right)+C$
6. $\frac{d}{d x}\left(\frac{1}{a} \csc ^{-1} \frac{x}{a}\right)=\frac{-1}{x \sqrt{x^2-a^2}} \Rightarrow \int \frac{-d x}{x \sqrt{x^2-a^2}}=\frac{1}{a} \csc ^{-1}\left(\frac{x}{a}\right)+C$

You can derive all the above results using the substitution method.

For example, to get the result (1), substitute  x = a sin Ө or x = a cos Ө and solve