Solve ∫ (from 1 to 2) of (x^3+5)^2(3x^2) wrt x

$$\int _ { 1 } ^ { 2 } ( x ^ { 3 } + 5 ) ^ { 2 } ( 3 x ^ { 2 } ) d x$$

Evaluate

$$\frac{1981}{3}\approx 660.333333333$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{3}+5\right)^{2}.

$$\int _{1}^{2}\left(\left(x^{3}\right)^{2}+10x^{3}+25\right)\times 3x^{2}\mathrm{d}x$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\int _{1}^{2}\left(x^{6}+10x^{3}+25\right)\times 3x^{2}\mathrm{d}x$$

Use the distributive property to multiply x^{6}+10x^{3}+25 by 3.

$$\int _{1}^{2}\left(3x^{6}+30x^{3}+75\right)x^{2}\mathrm{d}x$$

Use the distributive property to multiply 3x^{6}+30x^{3}+75 by x^{2}.

$$\int _{1}^{2}3x^{8}+30x^{5}+75x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 3x^{8}+30x^{5}+75x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3x^{8}\mathrm{d}x+\int 30x^{5}\mathrm{d}x+\int 75x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int x^{8}\mathrm{d}x+30\int x^{5}\mathrm{d}x+75\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{8}\mathrm{d}x with \frac{x^{9}}{9}. Multiply 3 times \frac{x^{9}}{9}.

$$\frac{x^{9}}{3}+30\int x^{5}\mathrm{d}x+75\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{5}\mathrm{d}x with \frac{x^{6}}{6}. Multiply 30 times \frac{x^{6}}{6}.

$$\frac{x^{9}}{3}+5x^{6}+75\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply 75 times \frac{x^{3}}{3}.

$$\frac{x^{9}}{3}+5x^{6}+25x^{3}$$

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.

$$25\times 2^{3}+5\times 2^{6}+\frac{2^{9}}{3}-\left(25\times 1^{3}+5\times 1^{6}+\frac{1^{9}}{3}\right)$$

Simplify.

$$\frac{1981}{3}$$