Solve ∫ (from 0 to 3) of (36t^3+36t^6+24t^2-84t^4-12t)dt

$$\int _ { 0 } ^ { 3 } ( 36 t ^ { 3 } + 36 t ^ { 6 } + 24 t ^ { 2 } - 84 t ^ { 4 } - 12 t ) d t$$

Evaluate

$$\frac{281961}{35}\approx 8056.028571429$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 36t^{3}+36t^{6}+24t^{2}-84t^{4}-12t\mathrm{d}t$$

Integrate the sum term by term.

$$\int 36t^{3}\mathrm{d}t+\int 36t^{6}\mathrm{d}t+\int 24t^{2}\mathrm{d}t+\int -84t^{4}\mathrm{d}t+\int -12t\mathrm{d}t$$

Factor out the constant in each of the terms.

$$36\int t^{3}\mathrm{d}t+36\int t^{6}\mathrm{d}t+24\int t^{2}\mathrm{d}t-84\int t^{4}\mathrm{d}t-12\int t\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{3}\mathrm{d}t with \frac{t^{4}}{4}. Multiply 36 times \frac{t^{4}}{4}.

$$9t^{4}+36\int t^{6}\mathrm{d}t+24\int t^{2}\mathrm{d}t-84\int t^{4}\mathrm{d}t-12\int t\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{6}\mathrm{d}t with \frac{t^{7}}{7}. Multiply 36 times \frac{t^{7}}{7}.

$$9t^{4}+\frac{36t^{7}}{7}+24\int t^{2}\mathrm{d}t-84\int t^{4}\mathrm{d}t-12\int t\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{2}\mathrm{d}t with \frac{t^{3}}{3}. Multiply 24 times \frac{t^{3}}{3}.

$$9t^{4}+\frac{36t^{7}}{7}+8t^{3}-84\int t^{4}\mathrm{d}t-12\int t\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{4}\mathrm{d}t with \frac{t^{5}}{5}. Multiply -84 times \frac{t^{5}}{5}.

$$9t^{4}+\frac{36t^{7}}{7}+8t^{3}-\frac{84t^{5}}{5}-12\int t\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t\mathrm{d}t with \frac{t^{2}}{2}. Multiply -12 times \frac{t^{2}}{2}.

$$9t^{4}+\frac{36t^{7}}{7}+8t^{3}-\frac{84t^{5}}{5}-6t^{2}$$

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.

$$9\times 3^{4}+\frac{36}{7}\times 3^{7}+8\times 3^{3}-\frac{84}{5}\times 3^{5}-6\times 3^{2}-\left(9\times 0^{4}+\frac{36}{7}\times 0^{7}+8\times 0^{3}-\frac{84}{5}\times 0^{5}-6\times 0^{2}\right)$$

Simplify.

$$\frac{281961}{35}$$