Solve ∫ (from 0 to 16) of (10a^{3}+2a^{2}b+12b^4)+(20a^3+3a^2b-15b^4)

$$\int _ { 0 } ^ { 16 } ( 10 a ^ { 3 } + 2 a ^ { 2 } b + 12 b ^ { 4 } ) + ( 20 a ^ { 3 } + 3 a ^ { 2 } b - 15 b ^ { 4 } )$$

Evaluate

$$480a^{3}-48b^{4}+80ba^{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 10a^{3}+2a^{2}b+12b^{4}+20a^{3}+3a^{2}b-15b^{4}\mathrm{d}x$$

Find the integral of 10a^{3}+2a^{2}b+12b^{4}+20a^{3}+3a^{2}b-15b^{4} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\left(10a^{3}+2a^{2}b+12b^{4}+20a^{3}+3a^{2}b-15b^{4}\right)x$$

Simplify.

$$\left(30a^{3}+5a^{2}b-3b^{4}\right)x$$

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.

$$\left(30a^{3}+5a^{2}b-3b^{4}\right)\times 16-\left(30a^{3}+5a^{2}b-3b^{4}\right)\times 0$$

Simplify.

$$480a^{3}+80a^{2}b-48b^{4}$$

Differentiate w.r.t. a

$$160a\left(9a+b\right)$$