\(Let\ 2x = t\)
\(∴ 2dx = dt\)
\(⇒ ∫\frac {1}{\sqrt {1+4x^2}} dx\)= \(\frac 12∫\frac {dt}{\sqrt {1+t^2}}\)
=\(\frac 12[log\ |t+\sqrt {t_2+1}|]+C\) \([∫\frac {1}{\sqrt {x^2+a^2}}dt = log\ |x+\sqrt {x^2+a^2}|]\)
=\(\frac 12 \ log\ |2x+\sqrt {4x^2+1}|+C\)
Let \( f : (0, \infty) \to \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0 \), } \[ \int_0^a f(x) \, dx = f(a), \quad f(1) = 1, \quad f(16) = \frac{1}{8}, \quad \text{then } 16 - f^{-1}\left( \frac{1}{16} \right) \text{ is equal to:}\]
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
These are tabulated below along with the meaning of each part.