Verified answer

You are doing good up to:

1/16 * [y*sin(y) + cos(y)] (evaluated from y=0 to π).

(It should read y=0 to π instead of x=0 to π, BTW!)

When you back-substituted for y = 4x, you needed to change the bounds back. You don't even need to back-substitute since you already changed the bounds!

From 1/16 * [y*sin(y) + cos(y)] (evaluated from y=0 to π), evaluate as usual to get:

1/16 * [y*sin(y) + cos(y)] (evaluated from y=0 to π)

= 1/16 * [(0 - 1) - (0 + 1)]

= -1/8.

I hope this helps!

Let's the Integration by Parts first.

Let u = x and dv = cos 4x dx

then du = dx and v = (1/4) sin 4x

(x/4) sin 4x - (1/4) ∫ sin 4x dx = (x/4) sin 4x + (1/16) cos 4x + C

Evaluating from x = 0 to x = π/4:

[(π/16) sin π + (1/16) cos π] - [(0) sin 0 + (1/16) cos 0] = 0 - 1/16 - 0 - 1/16 = -1/8

Let u = x and dv = cos(4x) dx

Then du = dx and v = (1/4)(sin 4x).

Thus by Integration by Parts

INT from 0 to pi/4 {x cos 4x} dx = (x/4) sin 4x - INT from 0 to pi/4 {(1/4) sin4x} dx

= [(x/4) sin 4x - (1/16) (-cos 4x)] evaulated at x = pi/4 and x = 0

= [(x/4) sin4x + (1/16) cos4x] evaulated at x = pi/4 and x = 0

= [((pi/4)/4)*sin(4*pi/4) + (1/16)*cos(4*pi/4)] - [(0/4)*sin(4*0) + (1/16)*cos(4*0)]

= [(pi/16)*sin pi + (1/16)*cos pi] - [0 + (1/16)*cos 0]

= [0 - 1/16] - [1/16]

= -1/8

(a million) Set t = cos(x) with a view to resolve ? tan(x) dx. So: t = cos(x) dt = -sin(x) dx ==> -dt = sin(x) dx we've that: ? tan(x) dx = ? sin(x) / cos(x) dx = ? - dt / t = -ln|t| + C = -ln|cos(x)| + C (2) utilising integration via factors, we get: ?[0,?/4] x/cos²x dx = ?[0,?/4] xsec²x dx u = x ; dv = sec²x dx du = dx ; v = tanx ? xsec²x dx = xtanx - ? tanx dx = xtan(x) + ln|cos(x)| Now use the bounds of integration from 0 to ?/4 to resolve thoroughly: (?/4)tan(?/4) + ln|cos(?/4)| - (0)tan(0) - ln|cos(0)| = (?/4) + ln(?(2)/2) - 0 - 0 = (?/4) + ln(?(2)/2)