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Math  /  Calculus

QuestionEvaluate each definite integral.
32. ∫−π/20cos⁡xdx\int_{-\pi / 2}^{0} \cos x d x

Studdy Solution

STEP 1

1. We are evaluating a definite integral of the function cos⁡x\cos x from −π/2-\pi/2 to 00.
2. The fundamental theorem of calculus will be used to evaluate the definite integral.

STEP 2

1. Find the antiderivative of cos⁡x\cos x.
2. Evaluate the antiderivative at the upper and lower limits of integration.
3. Subtract the value of the antiderivative at the lower limit from the value at the upper limit.

STEP 3

Find the antiderivative of cos⁡x\cos x. The antiderivative of cos⁡x\cos x is sin⁡x\sin x.
∫cos⁡x dx=sin⁡x+C \int \cos x \, dx = \sin x + C

STEP 4

Evaluate the antiderivative sin⁡x\sin x at the upper limit x=0x = 0 and at the lower limit x=−π/2x = -\pi/2.
sin⁡(0)=0 \sin(0) = 0 sin⁡(−π2)=−1 \sin\left(-\frac{\pi}{2}\right) = -1

STEP 5

Subtract the value of the antiderivative at the lower limit from the value at the upper limit:
∫−π/20cos⁡x dx=sin⁡(0)−sin⁡(−π2) \int_{-\pi/2}^{0} \cos x \, dx = \sin(0) - \sin\left(-\frac{\pi}{2}\right) =0−(−1) = 0 - (-1) =0+1 = 0 + 1 =1 = 1
The value of the definite integral is:
1 \boxed{1}

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