Integration I

Raise the power by one and divide by the new power. The constant of integration is compulsory.

$[x^2]_1^3 = 9 - 1 = 8$. Definite integrals need no $+C$, since it cancels in the subtraction.

Integrate term by term: $4x^3 \to x^4$, $-6x \to -3x^2$, $2 \to 2x$.

$[\sin x]_0^{\pi/2} = 1 - 0 = 1$. The limits are in radians.

$\left[4x - \dfrac{x^3}{3}\right]_{-2}^{2} = \left(8 - \dfrac83\right) - \left(-8 + \dfrac83\right) = \dfrac{32}{3}$. The curve is above the axis throughout.

Integrating sine gives $-\cos$, and the inner coefficient 3 requires a further division by 3. Check by differentiating.