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

The integral of 2xsin(x)+x^2cos(x) is x^2sin(x)+C

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integral of 2xsin(x)+x^2cos(x)

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

x^{2} sin (x) + C

Solution steps

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

\int 2 x sin (x) d x = 2 (- x cos (x) + sin (x))

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

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

Simplify

= x^{2} sin (x)

Add a constant to the solution

= x^{2} sin (x) + C

\int \frac{ln ( x )}{x + 4 x ln ^{2} ( x )} d x

\int \frac{- 2}{( x - 1 ) ^{3}} d x

\int - \frac{1}{x ( ln ( x ) ) ^{3}} d x

\int \frac{tan ^{3} ( x ) + 1}{cos ^{2} ( x )} d x

\int \frac{arctan ^{3} ( x )}{1 + x ^{2}} d x

What is the integral of 2xsin(x)+x^2cos(x) ?
The integral of 2xsin(x)+x^2cos(x) is x^2sin(x)+C