Brewer Reese
06/24/2024 · Junior High School
- \( \int \operatorname{sen} x \cos x d x \)
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Tutor-Verified Answer
Step-by-step Solution
Calculate the integral \( \int \sin x \cos x dx \).
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
\(\int \sin\left(x\right)\cos\left(x\right) dx\)
- step1: Use the substitution \(dx=-\frac{1}{\sin\left(x\right)} dt\) to transform the integral\(:\)
\(\int \sin\left(x\right)\cos\left(x\right)\left(-\frac{1}{\sin\left(x\right)}\right) dt\)
- step2: Simplify:
\(\int -\cos\left(x\right) dt\)
- step3: Use the substitution \(t=\cos\left(x\right)\) to transform the integral\(:\)
\(\int -t dt\)
- step4: Use properties of integrals:
\(-\int t dt\)
- step5: Evaluate the integral:
\(-\frac{1}{2}t^{2}\)
- step6: Substitute back:
\(-\frac{1}{2}\cos^{2}\left(x\right)\)
- step7: Add the constant of integral C:
\(-\frac{1}{2}\cos^{2}\left(x\right) + C, C \in \mathbb{R}\)
La integral de \( \int \sin x \cos x dx \) es igual a \( -\frac{1}{2}\cos^{2}(x) + C \), donde \( C \) es una constante real.
Quick Answer
La integral de \( \int \sin x \cos x dx \) es \( -\frac{1}{2}\cos^{2}(x) + C \).
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