Evaluating Trigonometric Definite Integral via Identity Substitution
Evaluate the definite integral \(\int_{0}^{\frac{\pi}{2}} \sin^2(x)\,dx\) using the identity \(\sin^2(x) = \frac{1-\cos(2x)}{2}\). Show all steps clearly.
Integration of Trigonometric Chain Integrals
Consider the integral of a trigonometric chain function of the form \(\int \cos^m(ax)\,dx\). Follow these steps: (a) Let \( u = ax \). Determine the expression for \( du \) in terms of \( dx \). (b) Rewrite the integral in terms of \( u \) and evaluate \( \int \cos^m(u)\,du \) using an appropriate […]
Reverse Chain Rule with sec^2 Function
Evaluate the indefinite integral \(\int \sec^2(kx)\,dx\) where k is a nonzero constant.
Integration with Trigonometric Substitution using Reverse Chain Rule
Evaluate the indefinite integral \(\int \cos x\,dx\) using the substitution \(u = \sin x\). Verify your answer by differentiation. Proceed by completing parts (a), (b), and (c).
Reverse Chain Rule: Trigonometric Composite Integration
Evaluate the integral \(\int 2x\sin(x^2)\,dx\).
Integration of Mixed Polynomial and Trigonometric Functions
Consider the function \(f(x) = 3x^2 + 4\sin(x)\). Break the integration into two parts: (a) Integrate the polynomial component \(3x^2\). (b) Integrate the trigonometric component \(4\sin(x)\).