Evaluate the Integral integral of 1-cos(x) with respect to x

Problem

(∫_^)(1−cos(x)*d(x))

Solution

1. Apply the sum rule for integration, which allows the integral of a difference to be split into the difference of two separate integrals.

(∫_^)(1−cos(x)*d(x))=(∫_^)(1*d(x))−(∫_^)(cos(x)*d(x))

2. Integrate the constant term 1 with respect to x which results in x

(∫_^)(1*d(x))=x

3. Integrate the trigonometric function cos(x) recalling that the derivative of sin(x) is cos(x)

(∫_^)(cos(x)*d(x))=sin(x)

4. Combine the results and add the constant of integration C

x−sin(x)+C

Final Answer

(∫_^)(1−cos(x)*d(x))=x−sin(x)+C

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