let-I-0-pi-x-cos-4-x-dxand-J-0-pi-x-sin-4-xdx-1-calculate-I-J-and-I-J-2-find-the-values-of-I-and-J-

Question Number 78264 by msup trace by abdo last updated on 15/Jan/20

l e t I = \int_{0}^{π} x {c o s}^{4} x d x a n d J = \int_{0}^{π} x {s i n}^{4} x d x

1) c a l c u l a t e I + J a n d I - J

2) f i n d t h e v a l u e s o f I a n d J

Commented by jagoll last updated on 15/Jan/20

I = \underset{0}{\int^{π}} x \cos^{4} x d x

l e t x = π - u {\begin{cases} x = 0 \Rightarrow u = π \\ x = π \Rightarrow u = 0 \end{cases}

I = \underset{π}{\int^{0}} (π - u) {\cos^{4} (π - u)} (- d u)

= \underset{0}{\int^{π}} π \cos^{4} u d u - I

2 I = \underset{0}{\int^{π}} π \cos^{4} x d x \Rightarrow I = \frac{π}{2} \underset{0}{\int^{π}} \cos^{4} x d x

Commented by jagoll last updated on 15/Jan/20

J = \underset{0}{\int^{π}} x \sin^{4} x d x . l e t x = π - u

J = \underset{0}{\int^{π}} π \sin^{4} x d x - J

J = \frac{π}{2} \underset{0}{\int^{π}} \sin^{4} x d x

Commented by jagoll last updated on 15/Jan/20

I + J = \frac{π}{2} \underset{0}{\int^{π}} \cos^{4} x + \sin^{4} x d x

= \frac{π}{2} \underset{0}{\int^{π}} (1 - \frac{1}{2} \sin^{2} (2 x)) d x

= \frac{π}{2} \underset{0}{\int^{π}} (\frac{1}{4} + \frac{1}{4} \cos (4 x)) d x

Commented by mathmax by abdo last updated on 18/Jan/20

1) w e h a v e I + J = \int_{0}^{π} x ({c o s}^{4} x + {s i n}^{4} x) d x

= \int_{0}^{π} x {{({c o s}^{2} x + {s i n}^{2} x)}^{2} - 2 {c o s}^{2} x {s i n}^{2} x} d x

= \int_{0}^{π} x {1 - 2 {(\frac{1}{2} s i n (2 x))}^{2}} d x = \int_{0}^{π} x {1 - \frac{1}{2} {s i n}^{2} (2 x)} d x

= \int_{0}^{π} x {1 - \frac{1}{2} \times \frac{1 - c o s (4 x)}{2}} d x = \int_{0}^{π} x {\frac{3}{4} + \frac{1}{4} c o s (4 x)} d x

= \frac{3}{4} \int_{0}^{π} x d x + \frac{1}{4} \int_{0}^{π} x c o s (4 x) d x

= \frac{3}{4} \times \frac{π^{2}}{2} + \frac{1}{4} {{[\frac{x}{4} s i n (4 x)]}_{0}^{π} - \int_{0}^{π} \frac{1}{4} s i n (4 x) d x}

= \frac{3 π^{2}}{8} - \frac{1}{16} {[- \frac{1}{4} c o s (4 x)]}_{0}^{π} = \frac{3 π^{2}}{8} a l s o w e h a v e

I - J = \int_{0}^{π} x ({c o s}^{4} x - {s i n}^{4} x) d x = \int_{0}^{π} x ({c o s}^{2} x - {s i n}^{2} x) d x

= \int_{0}^{π} x c o s (2 x) d x = {[\frac{x}{2} s i n (2 x)]}_{0}^{π} - \int_{0}^{π} \frac{1}{2} s i n (2 x) d x}

= - \frac{1}{2} {[- \frac{1}{2} c o s (2 x)]}_{0}^{π} = 0 \Rightarrow I = J \Rightarrow 2 I = \frac{3 π^{2}}{8} \Rightarrow I = \frac{3 π^{2}}{16} a n d

J = \frac{3 π^{2}}{16}