What is the integral of (4x+2)sin(x^2+x+2) ?

The integral of (4x+2)sin(x^2+x+2) is -2cos(x^2+x+2)+C

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integral of (4x+2)sin(x^2+x+2)

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

- 2 cos (x^{2} + x + 2) + C

Solution steps

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

Apply u-substitution

= \int 2 sin (u) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= 2 (- cos (u))

Substitute back

u = x^{2} + x + 2

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

Simplify

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

Add a constant to the solution

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

What is the integral of (4x+2)sin(x^2+x+2) ?
The integral of (4x+2)sin(x^2+x+2) is -2cos(x^2+x+2)+C