What is the integral of cos^6(x) ?

The integral of cos^6(x) is (sin(x)cos^5(x))/6+5/6 (1/4 cos^3(x)sin(x)+3/8 (x+1/2 sin(2x)))+C

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integral of cos^6(x)

\int cos^{6} (x) d x

\frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))) + C

Solution steps

\int cos^{6} (x) d x

Apply Integral Reduction

= \frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} \cdot \int cos^{4} (x) d x

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

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

Add a constant to the solution

= \frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))) + C

(x + \frac{y}{x}) d x - d y = 0

y^{^{'}} = 2 y + e^{2 x} cos (3 x), y (0) = 4

\frac{6 x ^{2} + 6 x + 6}{x}

\int sin (4 t) d t

\int sin (x) sin (2 x) d x

What is the integral of cos^6(x) ?
The integral of cos^6(x) is (sin(x)cos^5(x))/6+5/6 (1/4 cos^3(x)sin(x)+3/8 (x+1/2 sin(2x)))+C