What is the integral of x*cos^2(x) ?

The integral of x*cos^2(x) is 1/2 x(x+1/2 sin(2x))-1/2 ((x^2)/2-1/4 cos(2x))+C

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integral of x*cos^2(x)

\int x \cdot cos^{2} (x) d x

\frac{1}{2} x (x + \frac{1}{2} sin (2 x)) - \frac{1}{2} (\frac{x ^{2}}{2} - \frac{1}{4} cos (2 x)) + C

Solution steps

\int x cos^{2} (x) d x

Apply Integration By Parts

: \frac{1}{2} x (x + \frac{1}{2} sin (2 x)) - \int \frac{1}{2} (x + \frac{1}{2} sin (2 x)) d x

= \frac{1}{2} x (x + \frac{1}{2} sin (2 x)) - \int \frac{1}{2} (x + \frac{1}{2} sin (2 x)) d x

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

= \frac{1}{2} x (x + \frac{1}{2} sin (2 x)) - \frac{1}{2} (\frac{x ^{2}}{2} - \frac{1}{4} cos (2 x))

Add a constant to the solution

= \frac{1}{2} x (x + \frac{1}{2} sin (2 x)) - \frac{1}{2} (\frac{x ^{2}}{2} - \frac{1}{4} cos (2 x)) + C

g (x) = \frac{x}{x - 1}

f^{^{''}} (x) = \frac{6}{x}

x \to 0 lim (1 - 2 sin^{2} (\frac{x}{2}))

x \to 1 + lim (2 x - 2 - 2)

\int \frac{6 x + 5}{3 x ^{2} + 5 x} d x

What is the integral of x*cos^2(x) ?
The integral of x*cos^2(x) is 1/2 x(x+1/2 sin(2x))-1/2 ((x^2)/2-1/4 cos(2x))+C