Giải chi tiết 4.3. Tìm:,$a)~\int(3\sqrt x+\frac1{\sqrt[3]x})dx;$ $b)~\int\sqrt x(7x^2-3)dx(x0);$,$c)~\int\frac{(2x+1)^2}{x^2}dx;$ $d)~\int(2^x+\frac3{x^2})dx.$,4.4. Tìm:,$a)~\int(2\cos x-\frac3{\sin^2x})dx;$ $b)~\int4\sin^2\frac x2dx;$,$c)~\int(\sin\frac x2-\cos\frac x2)^2dx;$ $d)~\int(x+ an^2x)dx.$

Question

Giải chi tiết 4.3. Tìm:,$a)~\int(3\sqrt x+\frac1{\sqrt[3]x})dx;$ $b)~\int\sqrt x(7x^2-3)dx(x>0);$,$c)~\int\frac{(2x+1)^2}{x^2}dx;$ $d)~\int(2^x+\frac3{x^2})dx.$,4.4. Tìm:,$a)~\int(2\cos x-\frac3{\sin^2x})dx;$ $b)~\int4\sin^2\frac x2dx;$,$c)~\int(\sin\frac x2-\cos\frac x2)^2dx;$ $d)~\int(x+	an^2x)dx.$

Accepted Answer

4.3. Tìm:
a) $\int(3\sqrt{x} + \frac{1}{\sqrt[3]{x}})dx$

Ta có:
$$
\int(3\sqrt{x} + \frac{1}{\sqrt[3]{x}})dx = \int(3x^{1/2} + x^{-1/3})dx
$$
$$
= 3\int x^{1/2} dx + \int x^{-1/3} dx
$$
$$
= 3 \cdot \frac{x^{3/2}}{\frac{3}{2}} + \frac{x^{2/3}}{\frac{2}{3}} + C
$$
$$
= 2x^{3/2} + \frac{3}{2}x^{2/3} + C
$$

b) $\int \sqrt{x}(7x^2 - 3)dx$ (với $x > 0$)

Ta có:
$$
\int \sqrt{x}(7x^2 - 3)dx = \int (7x^{5/2} - 3x^{1/2})dx
$$
$$
= 7\int x^{5/2} dx - 3\int x^{1/2} dx
$$
$$
= 7 \cdot \frac{x^{7/2}}{\frac{7}{2}} - 3 \cdot \frac{x^{3/2}}{\frac{3}{2}} + C
$$
$$
= 2x^{7/2} - 2x^{3/2} + C
$$

c) $\int \frac{(2x+1)^2}{x^2}dx$

Ta có:
$$
\int \frac{(2x+1)^2}{x^2}dx = \int \frac{4x^2 + 4x + 1}{x^2}dx
$$
$$
= \int (4 + \frac{4}{x} + \frac{1}{x^2})dx
$$
$$
= 4\int dx + 4\int \frac{1}{x}dx + \int x^{-2}dx
$$
$$
= 4x + 4\ln|x| - \frac{1}{x} + C
$$

d) $\int(2^x + \frac{3}{x^2})dx$

Ta có:
$$
\int(2^x + \frac{3}{x^2})dx = \int 2^x dx + \int \frac{3}{x^2}dx
$$
$$
= \frac{2^x}{\ln 2} - \frac{3}{x} + C
$$

4.4. Tìm:
a) $\int(2\cos x - \frac{3}{\sin^2 x})dx$

Ta có:
$$
\int(2\cos x - \frac{3}{\sin^2 x})dx = 2\int \cos x dx - 3\int \csc^2 x dx
$$
$$
= 2\sin x + 3\cot x + C
$$

b) $\int 4\sin^2 \frac{x}{2} dx$

Ta có:
$$
\int 4\sin^2 \frac{x}{2} dx = 4\int \left(\frac{1 - \cos x}{2}\right)dx
$$
$$
= 2\int (1 - \cos x)dx
$$
$$
= 2\left(x - \sin x\right) + C
$$
$$
= 2x - 2\sin x + C
$$

c) $\int (\sin \frac{x}{2} - \cos \frac{x}{2})^2 dx$

Ta có:
$$
\int (\sin \frac{x}{2} - \cos \frac{x}{2})^2 dx = \int (\sin^2 \frac{x}{2} - 2\sin \frac{x}{2}\cos \frac{x}{2} + \cos^2 \frac{x}{2})dx
$$
$$
= \int (1 - \sin x)dx
$$
$$
= x + \cos x + C
$$

d) $\int (x + \tan^2 x)dx$

Ta có:
$$
\int (x + \tan^2 x)dx = \int x dx + \int \tan^2 x dx
$$
$$
= \frac{x^2}{2} + \int (\sec^2 x - 1)dx
$$
$$
= \frac{x^2}{2} + \tan x - x + C
$$
$$
= \frac{x^2}{2} + \tan x - x + C
$$

Đáp số:
4.3. 
a) $2x^{3/2} + \frac{3}{2}x^{2/3} + C$
b) $2x^{7/2} - 2x^{3/2} + C$
c) $4x + 4\ln|x| - \frac{1}{x} + C$
d) $\frac{2^x}{\ln 2} - \frac{3}{x} + C$

4.4. 
a) $2\sin x + 3\cot x + C$
b) $2x - 2\sin x + C$
c) $x + \cos x + C$
d) $\frac{x^2}{2} + \tan x - x + C$