Question

Помогите пожалуйста проинтегрировать уравнения?

Answer 1

$LARGE int frac{(4-x)^2}{4sqrt{x}}mathrm{d}x = int frac{16-8x+x^2}{4sqrt{x}}mathrm{d}x = int frac{16}{4sqrt{x}}mathrm{d}x+int frac{8x}{4sqrt{x}}mathrm{d}x+int frac{x^2}{4sqrt{x}}mathrm{d}x = 4int frac{mathrm{d}x}{sqrt{x}} +2int frac{x}{sqrt{x}}mathrm{d}x+frac{1}{4}int frac{x^2}{sqrt{x}}mathrm{d}x=8sqrt{x}+frac{4x^{frac{3}{2}}}{3}+frac{x^{frac{5}{2}}}{10}+C$

$LARGE int frac{5*3^x-4*2^x}{3^x}mathrm{d}x = 5int frac{3^x}{3^x}mathrm{d}x - 4int frac{2^x}{3^x}mathrm{d}x = 5int mathrm{d}x - 4int (frac{2}{3})^{x}mathrm{d}x = 5x-frac{(frac{2}{3})^{x}}{ln{frac{2}{3}}}+C$

$LARGE int frac{mathrm{d}x}{sqrt{2-2x^2}}=frac{1}{sqrt2}arcsinfrac{sqrt{2x}}{sqrt2}+C=frac{1}{sqrt2}arcsinsqrt{x}+C$