Tìm: a) nguyên hàm của (2x^5 +3) dx ; b) nguyên hàm (5cos x -3 sinx)dx

a) \(\int {\left( {2{x^5} + 3} \right)dx} \)\( = 2\int {{x^5}dx} + 3\int {dx} \)\( = \frac{{{x^6}}}{3} + 3x + C\).

b) \(\int {\left( {5\cos x - 3\sin x} \right)dx} \)\( = 5\int {\cos xdx} - 3\int {\sin xdx} \)\( = 5\sin x + 3\cos x + C\).

c) \(\int {\left( {\frac{{\sqrt x }}{2} - \frac{2}{x}} \right)} dx\)\( = \frac{1}{2}\int {{x^{\frac{1}{2}}}} dx - 2\int {\frac{1}{x}} dx\)\( = \frac{1}{3}{x^{\frac{3}{2}}} - 2\ln \left| x \right| + C\)\( = \frac{1}{3}x\sqrt x - 2\ln \left| x \right| + C\).

d) \(\int {\left( {{e^{x - 2}} - \frac{2}{{{{\sin }^2}x}}} \right)dx} \)\( = \frac{1}{{{e^2}}}\int {{e^x}dx} - 2\int {\frac{1}{{{{\sin }^2}x}}dx} \)\( = \frac{{{e^x}}}{{{e^2}}} + 2\cot x + C\)\( = {e^{x - 2}} + 2\cot x + C\).