Solve ∫ (5x^2-4x^1+2+pi) wrt x

$$\int ( 5 x ^ { 2 } - 4 x ^ { 1 + 2 } + \pi ) d x$$

Evaluate

$$-x^{4}+\frac{5x^{3}}{3}+\pi x+С$$

Solution Steps

Add 1 and 2 to get 3.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

$$\frac{5x^{3}}{3}-4\int x^{3}\mathrm{d}x+\int \pi \mathrm{d}x$$

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

$$\frac{5x^{3}}{3}-x^{4}+\int \pi \mathrm{d}x$$

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

$$\frac{5x^{3}}{3}-x^{4}+\pi x$$

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.

$$\frac{5x^{3}}{3}-x^{4}+\pi x+С$$

Differentiate w.r.t. x

$$\pi +5x^{2}-4x^{3}$$