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3.6 Q-11

Question Statement

Evaluate the integral: βˆ«Ο€6Ο€3cos⁑(t),dt\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \cos(t) , dt


Background and Explanation

To solve this integral, we need to recognize that the integral of cos⁑(t)\cos(t) is a standard antiderivative. Specifically: ∫cos⁑(t),dt=sin⁑(t)+C\int \cos(t) , dt = \sin(t) + C

Thus, we will integrate cos⁑(t)\cos(t) and then apply the limits of integration.


Solution

  1. Recognize the Standard Integral:

    The given integral is: βˆ«Ο€6Ο€3cos⁑(t),dt\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \cos(t) , dt

    Using the standard formula for the integral of cos⁑(t)\cos(t): ∫cos⁑(t),dt=sin⁑(t)\int \cos(t) , dt = \sin(t)

  2. Apply the Limits of Integration:

    Now, we evaluate sin⁑(t)\sin(t) at the limits Ο€3\frac{\pi}{3} and Ο€6\frac{\pi}{6}: sin⁑(t)βˆ£Ο€6Ο€3\sin(t) \bigg|_{\frac{\pi}{6}}^{\frac{\pi}{3}}

    This means we need to compute: sin⁑(Ο€3)βˆ’sin⁑(Ο€6)\sin\left(\frac{\pi}{3}\right) - \sin\left(\frac{\pi}{6}\right)

  3. Compute the Sine Values:

    From trigonometric values, we know: sin⁑(Ο€3)=32andsin⁑(Ο€6)=12\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \quad \text{and} \quad \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}

    Substituting these values, we get: 32βˆ’12\frac{\sqrt{3}}{2} - \frac{1}{2}

  4. Simplify the Result:

    Simplifying the expression: 3βˆ’12\frac{\sqrt{3} - 1}{2}

    Therefore, the value of the integral is: 3βˆ’12\boxed{\frac{\sqrt{3} - 1}{2}}


Key Formulas or Methods Used

  • Integral of Cosine:
    ∫cos⁑(t),dt=sin⁑(t)\int \cos(t) , dt = \sin(t)

Summary of Steps

  1. Recognize that ∫cos⁑(t),dt=sin⁑(t)\int \cos(t) , dt = \sin(t).

  2. Apply the limits of integration: sin⁑(Ο€3)βˆ’sin⁑(Ο€6)\sin\left(\frac{\pi}{3}\right) - \sin\left(\frac{\pi}{6}\right)

  3. Use known trigonometric values: sin⁑(Ο€3)=32,sin⁑(Ο€6)=12\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}, \quad \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}

  4. Simplify the result to: 3βˆ’12\frac{\sqrt{3} - 1}{2}

  5. Final answer: 3βˆ’12\boxed{\frac{\sqrt{3} - 1}{2}}