Evaluate Definite Integral of Cosine Over Square Root Function

Evaluate the integral $\displaystyle \int_{0}^{\pi/2} \frac{\cos t}{\sqrt{1 + \sin^2 t}} \, dt$.

In this problem, we are tasked with evaluating a definite integral involving a trigonometric function divided by the square root of an expression involving another trigonometric function. A clear strategy for tackling this type of integral includes recognizing and applying trigonometric identities and potentially substituting variables to transform the integral into a more solvable form. Understanding the relationship between the trigonometric functions involved is crucial. For instance, the function inside the integral involves both cosine and sine functions, which hints at potential simplifications or substitutions.

One approach is to consider trigonometric identities or substitutions that can simplify the integrand. For example, in problems like this, one might use a trigonometric substitution to express the sine in terms of the cosine or use identities to turn the integral into a standard form that is easier to integrate. The range of integration from zero to pi over two also indicates that the integral is over the first quadrant, where both sine and cosine functions are positive, which can simplify the boundaries and evaluation of the integral.

Another aspect to consider is checking if parts of the integrand suggest a standard form. If substitutions do not offer simplification, examining the need for integration techniques like trigonometric integrals or other integration strategies might be necessary. The key is to systematically simplify the integrand to a form that is manageable for computation, potentially transforming the original integral into a form that allows straightforward antiderivative calculation and subsequent evaluation at the given bounds.

Related Problems