"Solve the following integral:∫04πx|sin⁡x|dx1+|cos⁡x|" solved and explained

annlanw09y

Answered question

2022-03-23

Solve the following integral:

\int_{0}^{4 π} \frac{x | \sin x | d x}{1 + | \cos x |}

Leonardo Mcpherson

Beginner2022-03-24Added 13 answers

Let

I = \int_{0}^{π} d x \frac{x \sin x}{1 + | \cos x |}

and

J = \int_{0}^{π} d x \frac{\sin x}{1 + | \cos x |}

From a substitution of

x \mapsto π - x

, one may deduce that

I = (\frac{π}{2}) J

Now break up the interval

[0, 4 π]

into 4 equal pieces. In the k-th interval, we have

\int_{(k - 1) π}^{k π} d x \frac{x | \sin x |}{1 + | \cos x |}

There, make the sub

x = (k - 1) π + y

, and see that this integral is merely

I + (k - 1) π J

Thus the original integral is the sum of these integrals from

k = 1

4 I + 6 π J = 8 π J

. Thus, the problem reduces to evaluating J.

J = \int_{0}^{π} d x \frac{\sin x}{1 + | \cos x |} = 2 \int_{0}^{\frac{π}{2}} d x \frac{\sin x}{1 + \cos x} = 2 \log 2

Thus,

\int_{0}^{4 π} d x \frac{x | \sin x |}{1 + | \cos x |} = 16 π \log 2

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