Giả chi tiết giúp em ạ .

`a)`

`\int ( \frac{3}{\sqrt{x}} - \frac{5}{x} + 4e^x) dx`

`= \int \frac{3}{x^{1/2}} dx - \int \frac{5}{x} dx + \int 4e^x dx`

`=3 \int x^{-1/2} dx - 5 \int x^{-1} dx + 4 \int e^x dx`

`= 6\sqrt{x} - 5 \ln|x| + 4e^x + C`

`b)`

`\int \frac{2x^2 - 3x}{\sqrt[3]{x}} dx`

`=\int (2x^2 - 3x) . x^{-1/3} dx`

`=\int (2x^{5/3} - 3x^{2/3}) dx`

`=\frac{3}{4}x^{8/3} - \frac{9}{5}x^{5/3} + C`

`c)`

Đặt `u=1 - 3x^3 -> du = -9x^2 dx`

`\int x^2 \sin(1 - 3x^3) dx`

`->\int x^2 \sin(1 - 3x^3) dx`

`= -\frac{1}{9} \int \sin(u) du`

`= \frac{1}{9} \cos(u) + C = \frac{1}{9} \cos(1 - 3x^3) + C`

`d)`

Đặt : `u = x^2 + 2x -> u' = 2x + 2 `

`dv = \cos 2x dx -> v = \frac{1}{2} \sin 2x`

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`\int \cos 2x . (x^2 + 2x) dx`

`-> \frac{1}{2}(x^2 + 2x)\sin 2x - \frac{1}{2} \int (2x + 2)\sin 2x  dx`

`= \frac{1}{2}(x^2 + 2x)\sin 2x - \int x\sin 2x dx - \int \sin 2x dx`

`= \frac{1}{2}(x^2 + 2x)\sin 2x + \frac{1}{2}x \cos 2x - \frac{1}{4} \sin 2x + \frac{1}{2} \cos 2x + C`

`e)`

`\int \frac{dx}{\cos x}= \int \sec x  dx= \ln |\sec x + \tan x| + C`

`f)`

Đặt :

`t = \tan\frac{x}{2}-> dx = \frac{2dt}{1 + t^2}`

 và `\cos x = \frac{1 - t^2}{1 + t^2}`

`\int \frac{dx}{5 + 4\cos x}`

`=  \int \frac{\frac{2\,dt}{1 + t^2}}{5 + 4 .\frac{1 - t^2}{1 + t^2}}`

`=\int \frac{2\,dt}{9 + t^2}`

`= \frac{2}{3} \arctan( \frac{t}{3}) + C`

`-> \frac{2}{3} \arctan( \frac{\tan(x/2)}{3} ) + C`