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)\( \int (2 \sin x +3 \cos x) dx.\)
$$\int (2 \sin x +3 \cos x) dx = 2 \int \sin x dx + 3 \int \cos x dx = -2 \cos x + 3 \sin x+C. $$


2011-07-27 • Просмотров [ 3656 ]