Property 10
If integrand contains \(\sqrt{1-x^2}\) let \(x=\sin t\)
If integrand contains \(\sqrt{a^2-x^2}\) let \(x=a\cdot\sin t\)
If integrand contains \(1+x^2\) let \(x=\tan t\)
If integrand contains \(a^2+x^2\) let \(x=a\tan t\)
\(\blacksquare\quad\begin{aligned}\displaystyle\int\samefrac1{\sqrt{1-x^2}}\,dx&=\arcsin x+c\\&=-\arccos x+c\end{aligned}\)
\(\blacksquare\quad\begin{aligned}\displaystyle\int\samefrac1{1+x^2}\,dx&=\arctan x+c\\&=-\operatorname{arccot}x+c\end{aligned}\)
1.
\(\displaystyle\int \samefrac1{\sqrt{4-x^2}}\,dx\)
2.
\(\displaystyle\int \samefrac3{9+25x^2}\,dx=?\)
3.
\(\displaystyle\int \samefrac1{16+x^2}\,dx=?\)
4.
\(\displaystyle\int \samefrac1{9x^2+24x+20}\,dx=?\)
5.
\(\displaystyle\int \samefrac1{(x-7)^2+9}\,dx=?\)
6.
\(\displaystyle\int \samefrac1{\sqrt{20x-4x^2}}\,dx=?\)
7.
\(\displaystyle\int \samefrac x{1+x^4}\,dx=?\)
8.
\(\displaystyle\int \samefrac{2^x}{1+4^x}\,dx=?\)
9.
\(\displaystyle\int \samefrac7{\sqrt{1-49x^2}}\,dx=?\)
10.
\(\displaystyle\int \samefrac1{x^2+4x+13}\,dx=?\)