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

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

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integral of x(sin(x)-cos(x^2))

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

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

Solution steps

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

Expand

x (sin (x) - cos (x^{2})) : x sin (x) - x cos (x^{2})

= \int x sin (x) - x cos (x^{2}) d x

Apply the Sum Rule:

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

= \int x sin (x) d x - \int x cos (x^{2}) d x

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

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

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

Add a constant to the solution

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

y^{^{''}} + 4 y = 3 sin (x)

\frac{d V}{12 - 2 V} = d t, V (0) = 1

\frac{\partial}{\partial x} (\frac{x + 2 y}{3 x + 4 y})

\frac{d}{d θ} (cos (3 θ))

(1 + x + x^{2})^{99}

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