Solve ∫ (from - 1 to 1) of 6^4-y^2-5 wrt y

$$\int _ { - 1 } ^ { 1 } 6 ^ { 4 } - y ^ { 2 } - 5 d y$$

Evaluate

$$\frac{7744}{3}\approx 2581.333333333$$

Solution Steps

Calculate 6 to the power of 4 and get 1296.

$$\int _{-1}^{1}1296-y^{2}-5\mathrm{d}y$$

Subtract 5 from 1296 to get 1291.

$$\int _{-1}^{1}1291-y^{2}\mathrm{d}y$$

Evaluate the indefinite integral first.

$$\int 1291-y^{2}\mathrm{d}y$$

Integrate the sum term by term.

$$\int 1291\mathrm{d}y+\int -y^{2}\mathrm{d}y$$

Factor out the constant in each of the terms.

$$\int 1291\mathrm{d}y-\int y^{2}\mathrm{d}y$$

Find the integral of 1291 using the table of common integrals rule \int a\mathrm{d}y=ay.

$$1291y-\int y^{2}\mathrm{d}y$$

Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y^{2}\mathrm{d}y with \frac{y^{3}}{3}. Multiply -1 times \frac{y^{3}}{3}.

$$1291y-\frac{y^{3}}{3}$$

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.

$$1291\times 1-\frac{1^{3}}{3}-\left(1291\left(-1\right)-\frac{\left(-1\right)^{3}}{3}\right)$$

Simplify.

$$\frac{7744}{3}$$