Solve ∫ (5x-3)^2(2x-1) wrt x

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

Evaluate

$$\frac{25x^{4}}{2}-\frac{85x^{3}}{3}+24x^{2}-9x+С$$

Solution Steps

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

$$\int \left(25x^{2}-30x+9\right)\left(2x-1\right)\mathrm{d}x$$

Use the distributive property to multiply 25x^{2}-30x+9 by 2x-1 and combine like terms.

$$\int 50x^{3}-85x^{2}+48x-9\mathrm{d}x$$

Integrate the sum term by term.

$$\int 50x^{3}\mathrm{d}x+\int -85x^{2}\mathrm{d}x+\int 48x\mathrm{d}x+\int -9\mathrm{d}x$$

Factor out the constant in each of the terms.

$$50\int x^{3}\mathrm{d}x-85\int x^{2}\mathrm{d}x+48\int x\mathrm{d}x+\int -9\mathrm{d}x$$

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 50 times \frac{x^{4}}{4}.

$$\frac{25x^{4}}{2}-85\int x^{2}\mathrm{d}x+48\int x\mathrm{d}x+\int -9\mathrm{d}x$$

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 -85 times \frac{x^{3}}{3}.

$$\frac{25x^{4}}{2}-\frac{85x^{3}}{3}+48\int x\mathrm{d}x+\int -9\mathrm{d}x$$

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 48 times \frac{x^{2}}{2}.

$$\frac{25x^{4}}{2}-\frac{85x^{3}}{3}+24x^{2}+\int -9\mathrm{d}x$$

Find the integral of -9 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{25x^{4}}{2}-\frac{85x^{3}}{3}+24x^{2}-9x$$

Simplify.

$$24x^{2}-9x-\frac{85x^{3}}{3}+\frac{25x^{4}}{2}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$24x^{2}-9x-\frac{85x^{3}}{3}+\frac{25x^{4}}{2}+С$$

Differentiate w.r.t. x

$$\left(2x-1\right)\left(5x-3\right)^{2}$$