What is the integral of x(cos(x)-sin(x)) ?

The integral of x(cos(x)-sin(x)) is xsin(x)+xcos(x)+cos(x)-sin(x)+C

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integral of x(cos(x)-sin(x))

\int x (cos (x) - sin (x)) d x

x sin (x) + x cos (x) + cos (x) - sin (x) + C

Solution steps

\int x (cos (x) - sin (x)) d x

Apply Integration By Parts

: x (sin (x) + cos (x)) - \int sin (x) + cos (x) d x

= x (sin (x) + cos (x)) - \int sin (x) + cos (x) d x

\int sin (x) + cos (x) d x = - cos (x) + sin (x)

= x (sin (x) + cos (x)) - (- cos (x) + sin (x))

Simplify

x (sin (x) + cos (x)) - (- cos (x) + sin (x)) : x sin (x) + x cos (x) + cos (x) - sin (x)

= x sin (x) + x cos (x) + cos (x) - sin (x)

Add a constant to the solution

= x sin (x) + x cos (x) + cos (x) - sin (x) + C

\frac{d}{d x} (x - 0.8 - 0.2 \cdot sin (x))

\frac{d}{d x} (- \frac{( 2 x ^{2} + 1 )}{( 1 - x ^{2} ) ^{\frac{5}{2}}})

x \frac{d y}{d x} = y^{2}

sin (4 x)

\int - \frac{e ^{4 x}}{4 x} d x

What is the integral of x(cos(x)-sin(x)) ?
The integral of x(cos(x)-sin(x)) is xsin(x)+xcos(x)+cos(x)-sin(x)+C