Đáp án:

\(\begin{array}{l}
1)\quad A.\ \dfrac{4}{5}x^5 + C\\
2)\quad C.\ \dfrac23\sqrt{x^3} + C\\
3)\quad A.\ 2\sin\dfrac x2 + C\\
4)\quad B.\ \dfrac13\sqrt{x^3} + 4\sqrt x + C\\
5)\quad A.\ \dfrac{x^6}{6} - \dfrac{x^5}{5} + x^3 - \dfrac{3x^2}{2} + C\\
6)\quad D.\ \dfrac{x}{2} - \dfrac{\sin2x}{4} + C\\
7)\quad A.\ x^3 + \dfrac{x^2}{4} + C\\
8)\quad B.\ \dfrac{x^4}{2} - \dfrac{5x^2}{2} + 7x + C\\
9)\quad D.\ - \dfrac1x - \dfrac{x^3}{3} - \dfrac{x}{3} + C\\
10)\quad C.\ \dfrac32x^\tfrac23 + C\\
11)\quad A.\ -\dfrac{\cos3x}{3} + \dfrac{2x-\sin2x}{4} + C\\
12)\quad B.\ \displaystyle\int F'(x)dx = \dfrac83\sqrt{x^3} + \ln|x| -\dfrac83\\
13)\quad C.\ F(x)= 2\sin x + 3\cos x + \dfrac{x}{2} + \dfrac{\sin2x}{4} + C
\end{array}\) 

Giải thích các bước giải:

\(\begin{array}{l}
1)\quad I = \displaystyle\int 4x^4dx\\
\to I = \dfrac{4}{5}x^5 + C\\
2)\quad I = \displaystyle\int\sqrt xdx\\
\to I = \displaystyle\int x^{\tfrac12}dx\\
\to I = \dfrac{x^{\tfrac32}}{\dfrac32} + C\\
\to I = \dfrac23\sqrt{x^3} + C\\
3)\quad I = \displaystyle\int \cos\dfrac{x}{2}dx\\
\to I = 2\displaystyle\int \cos\dfrac{x}{2}\cdot \dfrac12dx\\
\to I = 2\displaystyle\int \cos\dfrac{x}{2}d\left(\dfrac x2\right)\\
\to I = 2\sin\dfrac x2 + C\\
4)\quad I = \displaystyle\int \left(\dfrac{\sqrt x}{2} + \dfrac{2}{\sqrt x}\right)dx\\
\to I = \displaystyle\int \dfrac{\sqrt x}{2}dx + \displaystyle\int\dfrac{2}{\sqrt x}dx\\
\to I = \dfrac12\displaystyle\int x^{\tfrac12}dx + 2\displaystyle\int x^{-\tfrac12}dx\\
\to I = \dfrac12\cdot \dfrac{x^{\tfrac32}}{\dfrac32} + 2\cdot \dfrac{x^{\tfrac12}}{\dfrac12} + C\\
\to I = \dfrac13\sqrt{x^3} + 4\sqrt x + C\\
5)\quad I = \displaystyle\int(x-1)(x^4 + 3x)dx\\
\to I = \displaystyle\int(x^5 - x^4 + 3x^2 - 3x)dx\\
\to I = \displaystyle\int x^5dx - \displaystyle\int x^4dx + 3\displaystyle\int x^2dx - 3\displaystyle\int xdx\\
\to I = \dfrac{x^6}{6} - \dfrac{x^5}{5} + x^3 - \dfrac{3x^2}{2} + C\\
6)\quad I = \displaystyle\int\sin^2xdx\\
\to I = \displaystyle\int\dfrac{1 -\cos2x}{2}dx\\
\to I = \dfrac12\displaystyle\int dx - \dfrac12\displaystyle\int\cos2xdx\\
\to I = \dfrac{x}{2} - \dfrac{\sin2x}{4} + C\\
7)\quad \displaystyle\int f(x)dx = \displaystyle\int \left(x^2 + \dfrac{x}{2}\right)dx\\
\to \displaystyle\int f(x)dx = 3\displaystyle\int x^2dx + \dfrac12\displaystyle\int xdx\\
\to \displaystyle\int f(x)dx = x^3 + \dfrac{x^2}{4} + C\\
8)\quad \displaystyle\int f(x)dx = \displaystyle\int (2x^3 - 5x + 7)dx\\
\to \displaystyle\int f(x)dx = 2\displaystyle\int x^3 - 5\displaystyle\int xdx + 7\displaystyle\int dx\\
\to \displaystyle\int f(x)dx = \dfrac{x^4}{2} - \dfrac{5x^2}{2} + 7x + C\\
9)\quad \displaystyle\int f(x)dx = \displaystyle\int\left(\dfrac{1}{x^2} - x^2 - \dfrac13\right)dx\\
\to \displaystyle\int f(x)dx = \displaystyle\int x^{-2}dx - \displaystyle\int x^2 - \dfrac13\displaystyle\int dx\\
\to \displaystyle\int f(x)dx= - \dfrac1x - \dfrac{x^3}{3} - \dfrac{x}{3} + C\\
10)\quad \displaystyle\int f(x)dx = \displaystyle\int x^{-\tfrac13}dx\\
\to \displaystyle\int f(x)dx = \dfrac{x^{\tfrac23}}{\dfrac23} + C\\
\to \displaystyle\int f(x)dx = \dfrac32x^\tfrac23 + C\\
11)\quad \displaystyle\int ydx = \displaystyle\int(\sin3x + \sin^2x)dx\\
\to \displaystyle\int ydx = \displaystyle\int \sin3xdx + \displaystyle\int\dfrac{1 - \cos2x}{2}dx\\
\to \displaystyle\int ydx = -\dfrac{\cos3x}{3} + \dfrac{x}{2}- \dfrac{\sin2x}{4} + C\\
\to \displaystyle\int ydx = -\dfrac{\cos3x}{3} + \dfrac{2x-\sin2x}{4} + C\\
12)\quad F'(x) = 4\sqrt x + \dfrac1x\\
\to \displaystyle\int F'(x)dx = \displaystyle\int\left(4\sqrt x + \dfrac1x\right)dx\\
\to \displaystyle\int F'(x)dx = 4\displaystyle\int x^{\tfrac12}dx + \displaystyle\int\dfrac1xdx\\
\to \displaystyle\int F'(x)dx = 4\cdot \dfrac{x^{\tfrac32}}{\dfrac32} + \ln|x| + C\\
\to \displaystyle\int F'(x)dx = \dfrac83\sqrt{x^3} + \ln|x| + C\\
\text{Ta lại có:}\\
\quad F(1) = 0\\
\to \displaystyle\int F'(x)dx = 0\\
\to \dfrac83\cdot \sqrt{1^3} + \ln 1 + C = 0\\
\to \dfrac83 + C = 0\\
\to C = -\dfrac83\\
\text{Vậy}\  \displaystyle\int F'(x)dx = \dfrac83\sqrt{x^3} + \ln|x|-\dfrac83\\
13)\quad F(x)= \displaystyle\int f(x)dx = \displaystyle\int(2\cos x -3\sin x + \cos^2x)dx\\
\to F(x)= 2\displaystyle\int\cos xdx - 3\displaystyle\int\sin xdx + \displaystyle\int\dfrac{1 + \cos2x}{2}dx\\
\to F(x)= 2\sin x + 3\cos x + \dfrac{x}{2} + \dfrac{\sin2x}{4} + C
\end{array}\)