What is the integral of (2x^3+2x)cos(x^2+1) ?

The integral of (2x^3+2x)cos(x^2+1) is x^2sin(x^2+1)+cos(x^2+1)+sin(x^2+1)+C

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integral of (2x^3+2x)cos(x^2+1)

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

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

Solution steps

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

Expand

(2 x^{3} + 2 x) cos (x^{2} + 1) : 2 x^{3} cos (x^{2} + 1) + 2 x cos (x^{2} + 1)

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

\int 2 x^{3} cos (x^{2} + 1) d x = x^{2} sin (x^{2} + 1) + cos (x^{2} + 1)

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

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

Add a constant to the solution

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

3 y y^{^{'}} = 7 x

8 + x - 7

y^{^{''}} - x = 0

x \to 1 lim (ln (x^{4} - 3 x + 10))

x \to - 3 + lim (\frac{x ^{2}}{x + 3})

What is the integral of (2x^3+2x)cos(x^2+1) ?
The integral of (2x^3+2x)cos(x^2+1) is x^2sin(x^2+1)+cos(x^2+1)+sin(x^2+1)+C