Solve ∫ (3x^5-6x^3+4x^4-4x^2-8) wrt x

$$\int ( 3 x ^ { 5 } - 6 x ^ { 3 } + 4 x ^ { 4 } - 4 x ^ { 2 } - 8 ) d x$$

Evaluate

$$\frac{x^{6}}{2}+\frac{4x^{5}}{5}-\frac{3x^{4}}{2}-\frac{4x^{3}}{3}-8x+С$$

Solution Steps

Integrate the sum term by term.

$$\int 3x^{5}\mathrm{d}x+\int -6x^{3}\mathrm{d}x+\int 4x^{4}\mathrm{d}x+\int -4x^{2}\mathrm{d}x+\int -8\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int x^{5}\mathrm{d}x-6\int x^{3}\mathrm{d}x+4\int x^{4}\mathrm{d}x-4\int x^{2}\mathrm{d}x+\int -8\mathrm{d}x$$

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 3 times \frac{x^{6}}{6}.

$$\frac{x^{6}}{2}-6\int x^{3}\mathrm{d}x+4\int x^{4}\mathrm{d}x-4\int x^{2}\mathrm{d}x+\int -8\mathrm{d}x$$

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 -6 times \frac{x^{4}}{4}.

$$\frac{x^{6}}{2}-\frac{3x^{4}}{2}+4\int x^{4}\mathrm{d}x-4\int x^{2}\mathrm{d}x+\int -8\mathrm{d}x$$

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

$$\frac{x^{6}}{2}-\frac{3x^{4}}{2}+\frac{4x^{5}}{5}-4\int x^{2}\mathrm{d}x+\int -8\mathrm{d}x$$

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 -4 times \frac{x^{3}}{3}.

$$\frac{x^{6}}{2}-\frac{3x^{4}}{2}+\frac{4x^{5}}{5}-\frac{4x^{3}}{3}+\int -8\mathrm{d}x$$

Find the integral of -8 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{6}}{2}-\frac{3x^{4}}{2}+\frac{4x^{5}}{5}-\frac{4x^{3}}{3}-8x$$

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.

$$\frac{x^{6}}{2}-\frac{3x^{4}}{2}+\frac{4x^{5}}{5}-\frac{4x^{3}}{3}-8x+С$$

Differentiate w.r.t. x

$$\left(x^{2}-2\right)\left(3x^{3}+4x^{2}+4\right)$$