another integration?

Question

cant figure this one out

integration [7x*ln(4x)*dx]

Answer 1

This will be an integration by parts.
Let u = ln(4x) and dv = 7x dx, then
du = 1/x dx and v = (7/2)x^2
integral of udv
= uv - integral of vdu
You take it from here.

Answer 2

Integral (7x ln(4x) dx)

The first thing you should do is pull out all constants. Constants sometimes get in the way of integration, and the wonderful thing about derivatives, integrals, and limits, is that we can ignore the constant when performing our desired function. In this case, our function is 7.

7 * Integral (x ln(4x) dx

The next thing we can do is recognize that ln(4x) = ln(4) + ln(x). This is true by one of the log properties.

7 * Integral ( x(ln4 + ln(x)) ) dx

Now, let's expand the inside.

7 * Integral ( xln(4) + xln(x) ) dx

With which we can change this into two separate integrals. Remember to distribute the 7 upon doing this.

7 * Integral (xln(4) dx) + 7*Integral (xln(x) dx)

Now, note that ln(4) is a constant, so we can pull it outside of the first integral.

7ln(4) * Integral (x dx) + 7*Integral (xln(x) dx)

The first integration is trivial. The second integration can be done using parts.

Let u = ln(x). dv = x dx
du = (1/x) dx. v = (1/2)x^2.

7ln(4) * (1/2)x^2 + 7[ (1/2)x^2 ln(x) - Integral ( (1/2) (1/x)x^2 dx) ]

This reduces to

(7/2)ln(4)x^2 + (7/2)x^2 ln(x) - (7/2) * Integral (x dx)

Which becomes

(7/2)ln(4)x^2 + (7/2) x^2 ln(x) - (7/2) (1/2)x^2 + C
(7/2)ln(4)x^2 + (7/2) x^2 ln(x) - (7/4) x^2 + C

Answer 3

Yup, purely integration. Radicals are lots much less perplexing once you write them out contained in this kind of powers. rather of coping with sq. roots, turn them into powers of a million/2. they are an identical element. for this reason, ?12x^3?x dx, it might purely be ?12x^(3.5 or 7/2) dx. combine from there. we could continually use the anti-power rule as a manner to try this, which says to function one to the exponent (we are gonna call that n), then divide each little thing by utilising n+a million. we are leaving the 12 the place that's, provided that's a relentless. This then leaves us 12x^(9/2) / 9/2, which all reduces to eight/3x^(9/2) + C. I extra the + C, for our integration consistent.

Answer 4

7 * x^2 * (2 * ln(4x) - 1) / 4

Answer 5

7x^2*ln(2)+(7/2)x^2*ln(x)- (7/4)x^2+C
or
(7/4)x^2 *(4ln(2)+2ln(x)-1) +C