Solve ∫ 25x^4*10x^2+1 wrt x

$$\int 25 x ^ { 4 } \cdot 10 x ^ { 2 } + 1 d x$$

Evaluate

$$\frac{250x^{7}}{7}+x+С$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\int 25x^{6}\times 10+1\mathrm{d}x$$

Multiply 25 and 10 to get 250.

$$\int 250x^{6}+1\mathrm{d}x$$

Integrate the sum term by term.

$$\int 250x^{6}\mathrm{d}x+\int 1\mathrm{d}x$$

Factor out the constant in each of the terms.

$$250\int x^{6}\mathrm{d}x+\int 1\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{6}\mathrm{d}x with \frac{x^{7}}{7}. Multiply 250 times \frac{x^{7}}{7}.

$$\frac{250x^{7}}{7}+\int 1\mathrm{d}x$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{250x^{7}}{7}+x$$

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.

$$\frac{250x^{7}}{7}+x+С$$

Differentiate w.r.t. x

$$250x^{6}+1$$