Question

Evaluate the equation

Answer 1

Step-by-step explanation:

We have,

$\rm \int2x \:cos^{2} (x) dx$

$\rm = \int x .2cos^{2} (x) dx$

$\rm = \int x \{1 + cos(2x) \} dx$

$\rm = \int x \: dx + \int \: x \: cos(2x) dx$

$\rm = \frac{ {x}^{2} }{2} + c _{1} + x\int cos(2x) dx - \int \bigg \{\frac{d}{dx}(x). \int \cos(2x) dx \bigg \} dx$

$\rm = \frac{ {x}^{2} }{2} + c _{1} + x. \frac{1}{2} . sin(2x) + c _{2} - \int \bigg \{1. \frac{1}{2} .sin(2x)\bigg \} dx$

Let $\rm\:c _{1} +c _{2} =k$

So,

$\rm = \frac{ {x}^{2} }{2} + \frac{x}{2} sin(2x) +k - \frac{1}{2} \int sin(2x) dx$

$\rm = \frac{ {x}^{2} }{2} + \frac{x}{2} sin(2x) +k - \frac{1}{2}. \frac{1}{2} . \{ - cos(2x) \} + c$

Let $\rm\:c +k=C$

$\rm = \frac{ {x}^{2} }{2} + \frac{x}{2} sin(2x) + \frac{1}{4} cos(2x) + C$