"Calculate the integral. ∫0π2xcos⁡2xdx" was answered by students like you

Suman Cole

Answered question

2021-10-26

Calculate the integral.

\int_{0}^{\frac{π}{2}} x \cos 2 x d x

cyhuddwyr9

Skilled2021-10-27Added 90 answers

Step 1
To evaluate the integral:

\int_{0}^{\frac{π}{2}} x \cos 2 x d x

Solution:
Integration by parts states that:

\int u v d x = u \int v d x - \int [\frac{d}{d x} (u) \int v d x] d x

Evaluating the integral using by parts,

\int_{0}^{\frac{π}{2}} x \cos 2 x d x = x \int_{0}^{\frac{π}{2}} \cos 2 x d x - \int_{0}^{\frac{π}{2}} [\frac{d}{d x} (x) \int \cos 2 x d x] d x

= {[x \cdot \frac{\sin 2 x}{2}]}_{0}^{\frac{π}{2}} - \int_{0}^{\frac{π}{2}} [1 \cdot \frac{\sin 2 x}{2}] d x

= \frac{1}{2} [\frac{π}{2} \sin (2 \cdot \frac{π}{2}) - 0] - \frac{1}{2} {[- \frac{\cos 2 x}{2}]}_{0}^{\frac{π}{2}}

= \frac{1}{2} [\frac{π}{2} \sin π] + \frac{1}{4} [\cos (2 \cdot \frac{π}{2}) - \cos 0]

= \frac{1}{2} \cdot 0 + \frac{1}{4} [\cos π - \cos 0]

= 0 + \frac{1}{4} [- 1 - 1]

= \frac{1}{4} \cdot (- 2)

= - \frac{1}{2}

Step 2
Hence, required answer is

- \frac{1}{2}

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