Solve ∫ (from 1 to 3) of 5x(x^2-7)^3 wrt x

$$\int _ { 1 } ^ { 3 } 5 x ( x ^ { 2 } - 7 ) ^ { 3 } d x$$

Evaluate

$$-800$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x^{2}-7\right)^{3}.

$$\int _{1}^{3}5x\left(\left(x^{2}\right)^{3}-21\left(x^{2}\right)^{2}+147x^{2}-343\right)\mathrm{d}x$$

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

$$\int _{1}^{3}5x\left(x^{6}-21\left(x^{2}\right)^{2}+147x^{2}-343\right)\mathrm{d}x$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\int _{1}^{3}5x\left(x^{6}-21x^{4}+147x^{2}-343\right)\mathrm{d}x$$

Use the distributive property to multiply 5x by x^{6}-21x^{4}+147x^{2}-343.

$$\int _{1}^{3}5x^{7}-105x^{5}+735x^{3}-1715x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 5x^{7}-105x^{5}+735x^{3}-1715x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 5x^{7}\mathrm{d}x+\int -105x^{5}\mathrm{d}x+\int 735x^{3}\mathrm{d}x+\int -1715x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$5\int x^{7}\mathrm{d}x-105\int x^{5}\mathrm{d}x+735\int x^{3}\mathrm{d}x-1715\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{7}\mathrm{d}x with \frac{x^{8}}{8}. Multiply 5 times \frac{x^{8}}{8}.

$$\frac{5x^{8}}{8}-105\int x^{5}\mathrm{d}x+735\int x^{3}\mathrm{d}x-1715\int x\mathrm{d}x$$

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 -105 times \frac{x^{6}}{6}.

$$\frac{5x^{8}}{8}-\frac{35x^{6}}{2}+735\int x^{3}\mathrm{d}x-1715\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply 735 times \frac{x^{4}}{4}.

$$\frac{5x^{8}}{8}-\frac{35x^{6}}{2}+\frac{735x^{4}}{4}-1715\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply -1715 times \frac{x^{2}}{2}.

$$\frac{5x^{8}}{8}-\frac{35x^{6}}{2}+\frac{735x^{4}}{4}-\frac{1715x^{2}}{2}$$

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.

$$\frac{5}{8}\times 3^{8}-\frac{35}{2}\times 3^{6}+\frac{735}{4}\times 3^{4}-\frac{1715}{2}\times 3^{2}-\left(\frac{5}{8}\times 1^{8}-\frac{35}{2}\times 1^{6}+\frac{735}{4}\times 1^{4}-\frac{1715}{2}\times 1^{2}\right)$$

Simplify.

$$-800$$