- \(\int {\dfrac{1}{{\sqrt {{a^2} - {x^2}} }}} \,dx = {\sin ^{ - 1}}\dfrac{x}{a} + c\)
- \(\int {\dfrac{{ - 1}}{{\sqrt {{a^2} - {x^2}} }}} \,dx = {\cos ^{ - 1}}\dfrac{x}{a} + c\)
- \(\int {\dfrac{1}{{{a^2} + {x^2}}}} \,dx = \dfrac{1}{a}{\tan ^{ - 1}}\dfrac{x}{a} + c\)
For Example: Find \(\int {\dfrac{1}{{\sqrt {4 - {x^2}} }}} \,dx\)\( = {\sin ^{ - 1}}\dfrac{x}{2} + c\)
For Example: Find \(\int {\dfrac{1}{{\sqrt {1 - 4{x^2}} }}} \,dx\)
\(\begin{align*}\int {\dfrac{1}{{\sqrt {1 - 4{x^2}} }}} \,dx &= \int {\dfrac{1}{{\sqrt {4\left( {{{\left( {1/2} \right)}^2} - {x^2}} \right)} }}} \,dx\\ &= \dfrac{1}{2}\int {\dfrac{1}{{\sqrt {{{\left( {1/2} \right)}^2} - {x^2}} }}} \,dx = \dfrac{1}{2}{\sin ^{ - 1}}2x\,\, + c\end{align*}\)
For Example: Find \(\int {\dfrac{1}{{9 + {x^2}}}} \,dx\)
\(\int {\dfrac{1}{{9 + {x^2}}}} \,dx = \dfrac{1}{3}{\tan ^{ - 1}}\dfrac{x}{3} + c\)
For Example: Find \(\int {\dfrac{1}{{1 + 9{x^2}}}} \,dx\)
\(\begin{align*}\int {\dfrac{1}{{1 + 9{x^2}}}} \,dx &= \int {\dfrac{1}{{9\left( {{{\left( {1/3} \right)}^2} + {x^2}} \right)}}} \,dx\\ &= \dfrac{1}{9} \times 3{\tan ^{ - 1}}3x\, = \dfrac{1}{3}{\tan ^{ - 1}}3x\,\, + c\end{align*}\)