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La integral es \( -\csc(x) + C \).
Solution
Calculate the integral \( \int \frac{\cos x}{\sin^{2} x} dx \).
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
\(\int \frac{\cos\left(x\right)}{\sin^{2}\left(x\right)} dx\)
- step1: Use the substitution \(dx=\frac{1}{\cos\left(x\right)} dt\) to transform the integral\(:\)
\(\int \frac{\cos\left(x\right)}{\sin^{2}\left(x\right)}\times \frac{1}{\cos\left(x\right)} dt\)
- step2: Simplify:
\(\int \frac{1}{\sin^{2}\left(x\right)} dt\)
- step3: Use the substitution \(t=\sin\left(x\right)\) to transform the integral\(:\)
\(\int \frac{1}{t^{2}} dt\)
- step4: Evaluate the integral:
\(\frac{t^{-2+1}}{-2+1}\)
- step5: Add the numbers:
\(\frac{t^{-1}}{-2+1}\)
- step6: Add the numbers:
\(\frac{t^{-1}}{-1}\)
- step7: Divide the terms:
\(-t^{-1}\)
- step8: Express with a positive exponent:
\(-\frac{1}{t}\)
- step9: Substitute back:
\(-\frac{1}{\sin\left(x\right)}\)
- step10: Rewrite the expression:
\(-\sin^{-1}\left(x\right)\)
- step11: Rewrite the expression:
\(-\csc\left(x\right)\)
- step12: Add the constant of integral C:
\(-\csc\left(x\right) + C, C \in \mathbb{R}\)
La integral de \( \frac{\cos x}{\sin^{2} x} \) con respecto a \( x \) es \( -\csc(x) + C \), donde \( C \) es una constante real.
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Simplify this solution Mind Expander
To solve the integral \( \int \frac{\cos x}{\sin^2 x} \, dx \), we can rewrite it using the fact that \( \frac{1}{\sin^2 x} = \csc^2 x \). Therefore, the integral becomes \( \int \cos x \cdot \csc^2 x \, dx \). Now, we can use substitution; let's set \( u = \sin x \). Then, \( du = \cos x \, dx \). This transforms the integral into \( \int \frac{1}{u^2} \, du \), which is easier to integrate: \[ \int u^{-2} \, du = -\frac{1}{u} + C = -\frac{1}{\sin x} + C. \] So the final answer is: \[ -\csc x + C. \]
