Solve ∫ (from 0 to 3) of (1+6w^2-10w^4)dw

$$\int _ { 0 } ^ { 3 } ( 1 + 6 w ^ { 2 } - 10 w ^ { 4 } ) d w$$

Evaluate

$$-429$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 1+6w^{2}-10w^{4}\mathrm{d}w$$

Integrate the sum term by term.

$$\int 1\mathrm{d}w+\int 6w^{2}\mathrm{d}w+\int -10w^{4}\mathrm{d}w$$

Factor out the constant in each of the terms.

$$\int 1\mathrm{d}w+6\int w^{2}\mathrm{d}w-10\int w^{4}\mathrm{d}w$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}w=aw.

$$w+6\int w^{2}\mathrm{d}w-10\int w^{4}\mathrm{d}w$$

Since \int w^{k}\mathrm{d}w=\frac{w^{k+1}}{k+1} for k\neq -1, replace \int w^{2}\mathrm{d}w with \frac{w^{3}}{3}. Multiply 6 times \frac{w^{3}}{3}.

$$w+2w^{3}-10\int w^{4}\mathrm{d}w$$

Since \int w^{k}\mathrm{d}w=\frac{w^{k+1}}{k+1} for k\neq -1, replace \int w^{4}\mathrm{d}w with \frac{w^{5}}{5}. Multiply -10 times \frac{w^{5}}{5}.

$$w+2w^{3}-2w^{5}$$

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.

$$3+2\times 3^{3}-2\times 3^{5}-\left(0+2\times 0^{3}-2\times 0^{5}\right)$$

Simplify.

$$-429$$