How do I proceed with this integral?

Question

$$ \int (2x)\ cos(5x)\ dx$$

I put

• $u = 2x$
• $du = 2\ dx$
• $v = \frac{1}{5}sin(5x)$
• $dv = cos(5x)\ dx$

Then I try $ uv - \int vdu $

$$ 2x \times \frac{1}{5}sin(5x) - \int 2\times\frac{1}{5}sin(5x)\ dx $$

which then gives me:

$$ 2x \times \frac{1}{5}sin(5x) + 2x \times \frac{1}{25}cos(5x) $$

This doesn't seem correct to me though.

Answer 1

Yes it's almost correct but the $x$ in second term in the final answer must not be there.

Answer must be $$2x \times \frac{1}{5}sin(5x) + 2\times \frac{1}{25}cos(5x)$$ Else everything is alright!