What is the integral of xsin(x)cos^3(x) ?

The integral of xsin(x)cos^3(x) is -1/4 xcos^4(x)+1/4 (cos^3(x)sin(x)+3/8 (x-1/4 sin(4x)))+C

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integral of xsin(x)cos^3(x)

integral

x sin (x) cos^{3} (x)

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

Solution steps

\int x sin (x) cos^{3} (x) d x

Apply Integration By Parts

: - \frac{1}{4} x cos^{4} (x) - \int - \frac{1}{4} cos^{4} (x) d x

= - \frac{1}{4} x cos^{4} (x) - \int - \frac{1}{4} cos^{4} (x) d x

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

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

Simplify

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

Add a constant to the solution

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

t^{3} (e^{t})^{2}

∣ x - 2 ∣ 30

\frac{1}{x ^{6}}

2 e^{u}

e^{3 x^{2}}

What is the integral of xsin(x)cos^3(x) ?
The integral of xsin(x)cos^3(x) is -1/4 xcos^4(x)+1/4 (cos^3(x)sin(x)+3/8 (x-1/4 sin(4x)))+C