Solve ∫ (from - 2 x to 1 - x ^ 2) of (1+r_{alpha})-(1-sqrt{2}) wrt y

$$\int _ { - 2 x } ^ { 1 - x ^ { 2 } } ( 1 + r _ { \alpha } ) - ( 1 - \sqrt { 2 } ) d y$$

Evaluate

$$\left(r_{α}+\sqrt{2}\right)\left(1+2x-x^{2}\right)$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 1+r_{α}-\left(1-\sqrt{2}\right)\mathrm{d}y$$

Find the integral of 1+r_{α}-\left(1-\sqrt{2}\right) using the table of common integrals rule \int a\mathrm{d}y=ay.

$$\left(1+r_{α}-\left(1-\sqrt{2}\right)\right)y$$

Simplify.

$$\left(r_{α}+\sqrt{2}\right)y$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\left(r_{α}+2^{\frac{1}{2}}\right)\left(1-x^{2}\right)-\left(r_{α}+2^{\frac{1}{2}}\right)\left(-2\right)x$$

Simplify.

$$\left(r_{α}+\sqrt{2}\right)\left(1-x^{2}+2x\right)$$

Differentiate w.r.t. x

$$2\left(1-x\right)\left(r_{α}+\sqrt{2}\right)$$