Integral of One Over Square Root of Quadratic Expression

Evaluate the integral of $\frac{1}{\sqrt{x^2 + 6x}}$ from 1 to 2.

Evaluating integrals of the form one over the square root of a quadratic expression often involves the technique of trigonometric substitution. This technique is helpful because it can simplify integrals that involve roots of quadratic expressions, which are otherwise challenging to integrate directly. In this problem, the expression under the square root is a quadratic in the form of x squared plus bx, which suggests a substitution related to trigonometric identities might be useful.

A strategic approach to solving such problems usually begins with completing the square for the quadratic expression under the square root. This can transform the integral into a form more conducive to trigonometric substitution. Typically, expressions like a squared minus x squared suggest sine substitution, while a squared plus x squared suggest tangent substitution. Once the substitution is made, the integral often becomes linear in terms of the trigonometric function, making it easier to integrate.

After performing the integration in terms of the trigonometric substitution, it is essential to translate the result back into the original variable. This often involves applying inverse trigonometric functions to resolve any trigonometric identities used during substitution. Careful attention to limits of integration, if applicable, or considering indefinite integrals, completes the problem-solving process.

Related Problems