What is the integral of ((x^3)/3)'sin(x) ?

The integral of ((x^3)/3)'sin(x) is -cos(x)((x^3)/3)'+2(xsin(x)+cos(x))+C

integral of ((x^3)/3)'sin(x)

\int (\frac{x ^{3}}{3})^{'} sin (x) d x

- cos (x) (\frac{x ^{3}}{3})' + 2 (x sin (x) + cos (x)) + C

Solution steps

\int (\frac{x ^{3}}{3})^{'} sin (x) d x

Apply Integration By Parts

= - cos (x) (\frac{x ^{3}}{3})^{^{'}} - \int - 2 x cos (x) d x

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

= - cos (x) (\frac{x ^{3}}{3})^{^{'}} - (- 2 (x sin (x) + cos (x)))

Simplify

= - cos (x) (\frac{x ^{3}}{3})^{^{'}} + 2 (x sin (x) + cos (x))

Add a constant to the solution

= - cos (x) (\frac{x ^{3}}{3})^{^{'}} + 2 (x sin (x) + cos (x)) + C

\frac{d}{d x} (\frac{2}{2 x - 1})

d er i v a t i v e 1 + sec (2 x)

l a pl a ce t r an s f or m cos (θ)

x \to 0 lim (e^{x})

d er i v a t i v e y = \frac{1}{x ^{2} + 1}

What is the integral of ((x^3)/3)'sin(x) ?
The integral of ((x^3)/3)'sin(x) is -cos(x)((x^3)/3)'+2(xsin(x)+cos(x))+C