1) \(\int (2x^3-5x^2+7x-3)dx.\)
$$\int (2x^3-5x^2+7x-3)dx=2 \int x^3dx-5\int x^2dx+7\int xdx-3\int dx=$$
$$=2 \cdot \frac{x^4}{4}-5\cdot \frac{x^3}{3}+7\cdot \frac{x^2}{2}-3x+C=$$
$$=\frac{1}{2}x^4-\frac{5}{3}x^3+\frac{7}{2}x^2-3x+C.$$
2) \(\int \left( \sqrt{x}+ \frac{1}{\sqrt[3]{x}}\right)^2dx.\)
$$\int \left( \sqrt{x}+ \frac{1}{\sqrt[3]{x}}\right)^2dx=\int \left(x+2 \cdot\frac{x^{1/2}}{x^{1/3}}+\frac{1}{x^{2/3}} \right)dx=$$
$$=\int (x+2x^{1/6}+x^{-2/3})dx=\int xdx+2\int x^{1/6}dx+\int x^{-2/3}dx=$$
$$=\frac{x^2}{2}+ 2\cdot \frac{x^{7/6}}{7/6}+\frac{x^{1/3}}{1/3}+C=\frac{x^2}{2}+\frac{12}{7}x\sqrt[6]{x}+3\sqrt[3]{x}+C.$$
3)\(\int 2^x \cdot 3^{2x}\cdot 5^{3x}dx.\)
$$\int 2^x \cdot 3^{2x}\cdot 5^{3x}dx =\int (2 \cdot 3^{2}\cdot 5^{3})^x dx=\frac{2250^x}{ \ln 2250}+C. $$
4)\( \int (1+x^2)^{1/2}xdx.\)
$$\int (1+x^2)^{1/2}xdx = \frac{1}{2} \int (1+x^2)^{1/2}\cdot 2xdx = \frac{1}{2} \int (1+x^2)^{1/2}d(1+x^2)= $$
$$= \frac{1}{2} \cdot \frac {(1+x^2)^{1/2+1}}{1/2+1}+C= \frac{1}{3}(1+x^2)^{3/2}+C.$$
5)\( \int (2 \sin x +3 \cos x) dx.\)
$$\int (2 \sin x +3 \cos x) dx = 2 \int \sin x dx + 3 \int \cos x dx = -2 \cos x + 3 \sin x+C. $$
