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Popular >

integral of sin^2(4y)cos^2(2y)

Solution

\int sin^{2} (4 y) cos^{2} (2 y) d y

Solution

2 (\frac{3}{4} y - \frac{3}{32} sin (8 y) + \frac{19}{24} cos^{3} (2 y) sin (2 y) - \frac{1}{6} cos^{5} (2 y) sin (2 y) - \frac{5}{32} (4 y + sin (4 y))) + C

Solution steps

\int sin^{2} (4 y) cos^{2} (2 y) d y

Apply Integral Substitution

= \int \frac{1}{2} sin^{2} (2 u) cos^{2} (u) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int sin^{2} (2 u) cos^{2} (u) d u

Rewrite using trig identities

= \frac{1}{2} \cdot \int 4 cos^{4} (u) sin^{2} (u) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot 4 \cdot \int cos^{4} (u) sin^{2} (u) d u

Rewrite using trig identities

= \frac{1}{2} \cdot 4 \cdot \int cos^{4} (u) (1 - cos^{2} (u)) d u

Expand

cos^{4} (u) (1 - cos^{2} (u)) : cos^{4} (u) - cos^{6} (u)

= \frac{1}{2} \cdot 4 \cdot \int cos^{4} (u) - cos^{6} (u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} \cdot 4 (\int cos^{4} (u) d u - \int cos^{6} (u) d u)

\int cos^{4} (u) d u = cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 u))

\int cos^{6} (u) d u = \frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u)))

= \frac{1}{2} \cdot 4 (cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 u)) - (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u)))))

Substitute back

u = 2 y

= \frac{1}{2} \cdot 4 (cos^{3} (2 y) sin (2 y) + \frac{3}{8} (2 y - \frac{1}{4} sin (4 \cdot 2 y)) - (\frac{sin ( 2 y ) cos ^{5} ( 2 y )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (2 y) sin (2 y) + \frac{3}{8} (2 y + \frac{1}{2} sin (2 \cdot 2 y)))))

Simplify

\frac{1}{2} \cdot 4 (cos^{3} (2 y) sin (2 y) + \frac{3}{8} (2 y - \frac{1}{4} sin (4 \cdot 2 y)) - (\frac{sin ( 2 y ) cos ^{5} ( 2 y )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (2 y) sin (2 y) + \frac{3}{8} (2 y + \frac{1}{2} sin (2 \cdot 2 y))))) : 2 (\frac{3}{4} y - \frac{3}{32} sin (8 y) + \frac{19}{24} cos^{3} (2 y) sin (2 y) - \frac{1}{6} cos^{5} (2 y) sin (2 y) - \frac{5}{32} (4 y + sin (4 y)))

= 2 (\frac{3}{4} y - \frac{3}{32} sin (8 y) + \frac{19}{24} cos^{3} (2 y) sin (2 y) - \frac{1}{6} cos^{5} (2 y) sin (2 y) - \frac{5}{32} (4 y + sin (4 y)))

Add a constant to the solution

= 2 (\frac{3}{4} y - \frac{3}{32} sin (8 y) + \frac{19}{24} cos^{3} (2 y) sin (2 y) - \frac{1}{6} cos^{5} (2 y) sin (2 y) - \frac{5}{32} (4 y + sin (4 y))) + C

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