n.karthick said:You can refer to the standard integrals in the PF library to get the solution directly.
U-substitution is a method used in integration to simplify an integral by substituting a new variable for the original variable. This allows us to rewrite the integral in terms of the new variable, making it easier to solve.
In this case, u-substitution is useful because it allows us to simplify the integrand by substituting $\sqrt{9x^{2}+4}$ with a new variable, making it easier to integrate.
The best choice for u-substitution is to let $u=3x$, which will result in $du=3dx$, allowing us to rewrite the integral as $\frac{1}{3}\int\sqrt{u^{2}+4}du$. This simplifies the integral and makes it easier to solve.
No, u-substitution is not the only method for integrating $\sqrt{9x^{2}+4}$. Other methods such as integration by parts or trigonometric substitutions can also be used. However, in most cases, u-substitution is the simplest method.
One common mistake is forgetting to substitute the new variable back into the final answer. It is important to remember to replace all instances of the new variable in the final answer with the original variable. Another mistake is choosing an incorrect u-substitution, which can lead to a more complex integral. It is important to carefully choose the substitution that will simplify the integral the most.