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Math  /  Calculus

QuestionFind the definite integral. (Use symbolic notation and fractions where needed.) ∫−π/4π/2cos⁡(x)dx=\int_{-\pi / 4}^{\pi / 2} \cos (x) d x= □\square

Studdy Solution

STEP 1

1. We are tasked with finding the definite integral of the function cos⁡(x) \cos(x) over the interval [−π/4,π/2][- \pi / 4, \pi / 2].

STEP 2

1. Find the indefinite integral of cos⁡(x) \cos(x) .
2. Evaluate the indefinite integral at the upper limit of the interval.
3. Evaluate the indefinite integral at the lower limit of the interval.
4. Subtract the value found at the lower limit from the value found at the upper limit to find the definite integral.

STEP 3

Find the indefinite integral of cos⁡(x) \cos(x) :
The indefinite integral of cos⁡(x) \cos(x) is:
∫cos⁡(x) dx=sin⁡(x)+C \int \cos(x) \, dx = \sin(x) + C
where C C is the constant of integration.

STEP 4

Evaluate the indefinite integral at the upper limit π/2 \pi / 2 :
sin⁡(π2)=1 \sin\left(\frac{\pi}{2}\right) = 1

STEP 5

Evaluate the indefinite integral at the lower limit −π/4-\pi / 4:
sin⁡(−π4)=−22 \sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}

STEP 6

Subtract the value found at the lower limit from the value found at the upper limit:
∫−π/4π/2cos⁡(x) dx=1−(−22) \int_{-\pi / 4}^{\pi / 2} \cos(x) \, dx = 1 - \left(-\frac{\sqrt{2}}{2}\right)
=1+22 = 1 + \frac{\sqrt{2}}{2}
The value of the definite integral is:
1+22 \boxed{1 + \frac{\sqrt{2}}{2}}

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