Integral Of Cos 2x Sinx: The Trick That Simplifies It
Integral of cos 2x sin x: Why Identities Matter Here
The integral $$\int \cos(2x)\sin x\,dx$$ resolves to a precise antiderivative using trigonometric identities, yielding a result that is both elegant and practical for educational leadership applications in Marist education. The primary query is answered: the integral evaluates to $$-\frac{1}{4}\cos x - \frac{1}{12}\cos 3x + C$$. This result emerges from a systematic use of product-to-sum identities or, equivalently, expressing $$\cos(2x)$$ in terms of $$\cos x$$ and applying standard integration techniques. Understanding this calculation reinforces rigorous math pedagogy we advocate for in Catholic and Marist schools across Brazil and Latin America, where mathematical literacy supports evidence-based curriculum design and student achievement.
Derivation Overview
One concise pathway starts with a product-to-sum identity: $$\cos(2x)\sin x = \tfrac{1}{2}[\sin(3x) - \sin x]$$. Integrating term-by-term gives a straightforward antiderivative: $$\int \cos(2x)\sin x\,dx = \tfrac{1}{2}\int \sin(3x)\,dx - \tfrac{1}{2}\int \sin x\,dx = -\tfrac{1}{6}\cos(3x) + \tfrac{1}{2}\cos x + C$$. After simplification, this is equivalent to $$-\frac{1}{4}\cos x - \frac{1}{12}\cos(3x) + C$$. This linkage between identities and integration showcases precise mathematical reasoning-a value we emphasize in Marist pedagogy as a model for disciplined inquiry.
Alternative Path: Substitution Route
Alternatively, rewrite $$\cos(2x)$$ as $$1 - 2\sin^2 x$$ and proceed with substitution. Let $$u=\sin x$$, $$du=\cos x\,dx$$. The integral becomes $$\int (1-2u^2)\,du$$ after adjusting for the differential, leading to the same antiderivative up to a constant. This dual approach demonstrates how multiple mathematical tools converge on a singular correct result, mirroring how diverse educational strategies converge on holistic student outcomes in Marist schools.
Key Takeaways for Educators
- Identity importance: Product-to-sum identities unlock otherwise complex integrals and illuminate the interconnectedness of trigonometric relationships.
- Method flexibility: Different pathways yield the same answer,Providing teachers with robust explanations for varied student learning styles.
- Practice value: Regular exposure to such problems strengthens procedural fluency and conceptual understanding in the mathematics curriculum we advocate.
- State the integrand: $$\cos(2x)\sin x$$.
- Choose a viable identity: $$\cos(2x)\sin x = \tfrac{1}{2}[\sin(3x) - \sin x]$$.
- Integrate term-by-term: $$-\frac{1}{6}\cos(3x) + \frac{1}{2}\cos x + C$$.
- Optionally simplify to the equivalent form: $$-\frac{1}{4}\cos x - \frac{1}{12}\cos(3x) + C$$.
Illustrative Data Table
| Step | Description | Result |
|---|---|---|
| 1 | Apply product-to-sum identity to the integrand | $$\tfrac{1}{2}[\sin(3x) - \sin x]$$ |
| 2 | Integrate each term separately | $$-\tfrac{1}{6}\cos(3x) + \tfrac{1}{2}\cos x$$ |
| 3 | Simplify expression | $$-\tfrac{1}{4}\cos x - \tfrac{1}{12}\cos(3x) + C$$ |
Frequently Asked Questions
Helpful tips and tricks for Integral Of Cos 2x Sinx The Trick That Simplifies It
[Answer]?
The antiderivative is $$-\frac{1}{4}\cos x - \frac{1}{12}\cos(3x) + C$$. This result can be obtained via the product-to-sum identity or by substitution, illustrating the consistency of trigonometric methods in integral calculus.
[Answer]?
Product-to-sum identities simplify products of trig functions into sums, enabling straightforward integration term-by-term. This approach highlights how identities support efficient problem-solving in calculus, a skill we champion in Marist mathematics education.
[Answer]?
Yes. Each follows from appropriate identities (product-to-sum or angle addition formulas), leading to solvable integrals and reinforcing a consistent, disciplined approach to trigonometric integration essential for student proficiency in STEM tracks.
