Solve ∫ _{-pi/2}^frac{pi{2}}(2g^2+siny) wrt y=

$$\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } ( 2 g ^ { 2 } + \sin y ) d y =$$

Evaluate

$$2\pi g^{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 2g^{2}+\sin(y)\mathrm{d}y$$

Integrate the sum term by term.

$$\int 2g^{2}\mathrm{d}y+\int \sin(y)\mathrm{d}y$$

Factor out the constant in each of the terms.

$$2\int g^{2}\mathrm{d}y+\int \sin(y)\mathrm{d}y$$

Find the integral of g^{2} using the table of common integrals rule \int a\mathrm{d}y=ay.

$$2g^{2}y+\int \sin(y)\mathrm{d}y$$

Use \int \sin(g)\mathrm{d}g=-\cos(g) from the table of common integrals to obtain the result.

$$2g^{2}y-\cos(y)$$

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.

$$\frac{1}{2}\times 2g^{2}\pi -\cos(\frac{1}{2}\pi )-\left(-\frac{1}{2}\times 2g^{2}\pi -\cos(-\frac{1}{2}\pi )\right)$$

Simplify.

$$2g^{2}\pi$$

Differentiate w.r.t. g

$$4\pi g$$