Dựa vào các đáp án, xét:

$\mathop \smallint \limits_{ - 1}^2 f(2x)dx = \frac{1}{2}\mathop \smallint \limits_{ - 1}^2 f(2x)d(2x) = \frac{1}{2}\mathop \smallint \limits_{ - 2}^4 f(x)dx = 1$

$\begin{array}{*{20}{l}}{\mathop \smallint \limits_{ - 3}^3 f(x + 1)dx = \mathop \smallint \limits_{ - 3}^3 f(x + 1)d(x + 1)}\\{ = \mathop \smallint \limits_{ - 2}^4 f(x)dx = 2}\end{array}$

$\mathop \smallint \limits_0^6 \frac{1}{2}f(x - 2)dx = \mathop \smallint \limits_0^6 \frac{1}{2}f(x - 2)d(x - 2) = \frac{1}{2}\mathop \smallint \limits_{ - 2}^4 f(x)dx = 1$

Do đó các đáp án B, C, D đều đúng, đáp án A sai.

Hàm số$y = f\left( x \right)$là hàm số lẻ nếu$f\left( x \right) = - f\left( { - x} \right)$

Đặt$x = - t \Rightarrow dx = - dt$

Đổi cận$\left\{ {\begin{array}{*{20}{c}}{x = a \Rightarrow t = - a}\\{x = - a \Rightarrow t = a}\end{array}} \right.$

$\Rightarrow \mathop \smallint \limits_{ - a}^a f\left( x \right)dx = \mathop \smallint \limits_a^{ - a} f\left( { - t} \right)\left( { - dt} \right) = \mathop \smallint \limits_{ - a}^a \left( { - f\left( t \right)} \right)dt = - \mathop \smallint \limits_{ - a}^a f\left( t \right)dt = - \mathop \smallint \limits_{ - a}^a f\left( x \right)dx$

Do đó

$\mathop \smallint \limits_{ - a}^a f\left( x \right)dx = - \mathop \smallint \limits_{ - a}^a f\left( x \right)dx \Leftrightarrow 2\mathop \smallint \limits_{ - a}^a f\left( x \right)dx = 0 \Leftrightarrow \mathop \smallint \limits_{ - a}^a f\left( x \right)dx = 0$

Đặt$4x = t$khi đó$4dx = dt$

Đổi cận với x=0 thì t=0;x=1 thì t=4

$\mathop \smallint \limits_0^1 f\left( {4x} \right)dx = \frac{1}{4}\mathop \smallint \limits_0^4 f(t)dt = - \frac{1}{4}$

vì tích phân không phụ thuộc vào biến số.

Đặt$u = 8 + \cos x \Rightarrow du = - \sin xdx \Rightarrow \sin xdx = - du$

Đổi cận:$\left\{ {\begin{array}{*{20}{c}}{x = 0 \Rightarrow t = 9}\\{x = \frac{\pi }{2} \Rightarrow t = 8}\end{array}} \right. \Rightarrow I = - \mathop \smallint \limits_9^8 \sqrt u du = \mathop \smallint \limits_8^9 \sqrt u du$

Đặt$u = \sqrt {{e^x} - 1} \Rightarrow {u^2} = {e^x} - 1 \Rightarrow 2udu = {e^x}dx$ và${e^x} = {u^2} + 1$

Đổi cận:$\left\{ {\begin{array}{*{20}{c}}{x = ln2 \Rightarrow u = 1}\\{x = ln5 \Rightarrow u = 2}\end{array}} \right.$

Khi đó ta có

$I = \int\limits_{\ln 2}^{\ln 5} {\frac{{{e^{2x}}}}{{\sqrt {{e^x} - 1} }}dx = 2\int\limits_1^2 {\frac{{({u^2} + 1)udu}}{u}} } = 2\int\limits_1^2 {({u^2} + 1)du} = = 2\left( {\frac{{{u^3}}}{3} + u} \right)\left| {_1^2} \right.$

Đặt${x^2} + 1 = t \Rightarrow 2xdx = dt \Rightarrow xdx = \frac{{dt}}{2}$

Đổi cận$\left\{ {\begin{array}{*{20}{c}}{x = 0 \Rightarrow t = 1}\\{x = 1 \Rightarrow t = 2}\end{array}} \right.$

Khi đó ta có:

$I = \int\limits_0^1 {\frac{x}{{{x^2} + 1}}} dx = \frac{1}{2}\int\limits_1^2 {\frac{{dt}}{t}} = \frac{1}{2}\ln \left| t \right|\left| {_1^2} \right. = \frac{1}{2}(ln2 - ln1)$

$= \frac{1}{2}\ln 2 = \ln \sqrt 2 \Rightarrow a = \sqrt 2 \,\,\left( {tm} \right)$

Đặt u = lnx ⇒$\Rightarrow du = \frac{{dx}}{x}$ và$x = {e^u}$

Đổi cận: $\left\{ {\begin{array}{*{20}{c}}{x = 1 \Rightarrow u = 0}\\{x = e \Rightarrow u = 1}\end{array}} \right.$

Khi đó ta có: $I = \mathop \smallint \limits_1^e \frac{{1 - \ln x}}{{{x^2}}}dx = \mathop \smallint \limits_0^1 \frac{{1 - u}}{{{e^u}}}du = \mathop \smallint \limits_0^1 \left( {1 - u} \right){e^{ - u}}du$

Đặt

$t = \sqrt {1 + 3\ln x} \Rightarrow {t^2} = 1 + 3\ln x \Rightarrow 2tdt = \frac{{3dx}}{x} \Rightarrow \frac{{dx}}{x} = \frac{2}{3}tdt$

Đổi cận: $\left\{ {\begin{array}{*{20}{c}}{x = 1 \Rightarrow t = 1}\\{x = e \Rightarrow t = 2}\end{array}} \right.$

Khi đó ta có:

$I = \frac{2}{3}\int\limits_1^2 {{t^2}dt = } \frac{2}{3}\frac{{{t^3}}}{3}\left| {_1^2} \right. = \frac{2}{9}{t^3}\left| {_1^2} \right. = \left( {\frac{2}{9}{t^3} + 2} \right)\left| {_1^2} \right. = \frac{2}{9}(8 - 1) = \frac{{14}}{9}$

Vậy A sai.

Cách 1: Đặt$t = {\ln ^2}x + 1 \Rightarrow dt = 2\ln x\frac{{dx}}{x} \Rightarrow \frac{{\ln xdx}}{x} = \frac{{dt}}{2}$

Đổi cận:$\left\{ {\begin{array}{*{20}{c}}{x = 1 \Rightarrow t = 1}\\{x = e \Rightarrow t = 2}\end{array}} \right.$

Khi đó ta có:

$I = \frac{1}{2}\int\limits_1^2 {\frac{{dt}}{t}} = \frac{1}{2}\ln \left| t \right|\left| {_1^2} \right. = \frac{1}{2}\ln 2 = aln2 + b \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{a = \frac{1}{2}}\\{b = 0}\end{array}} \right. \Rightarrow 2a + b = 1$

Đặt

$\begin{array}{*{20}{l}}{t = \frac{{\sqrt {{x^2} + 1} }}{x} \Leftrightarrow {t^2} = \frac{{{x^2} + 1}}{{{x^2}}} = 1 + \frac{1}{{{x^2}}}}\\{ \Rightarrow 2tdt = - \frac{2}{{{x^3}}}dx \Rightarrow tdt = - \frac{{dx}}{{{x^3}}}}\end{array}$

Và${t^2}{x^2} = {x^2} + 1 \Rightarrow {x^2}\left( {{t^2} - 1} \right) = 1 \Leftrightarrow {x^2} = \frac{1}{{{t^2} - 1}} \Rightarrow \frac{{dx}}{x} = - \frac{t}{{{t^2} - 1}}dt$

Đổi cận:$\left\{ {\begin{array}{*{20}{c}}{x = 1 \Rightarrow t = \sqrt 2 }\\{x = \sqrt 3 \Rightarrow t = \frac{2}{{\sqrt 3 }}}\end{array}} \right.$

Khi đó ta có:$I = - \mathop \smallint \limits_{\sqrt 2 }^{\frac{2}{{\sqrt 3 }}} \frac{{{t^2}}}{{{t^2} - 1}}dt$

Đặt$x = 4\sin t \Rightarrow dx = 4\cos tdt$

Đổi cận: $\left\{ {\begin{array}{*{20}{c}}{x = 0 \Rightarrow t = 0}\\{x = \sqrt 8 \Rightarrow t = \frac{\pi }{4}}\end{array}} \right.$

Khi đó ta có:

$I = 4\mathop \smallint \limits_0^{\frac{\pi }{4}} \sqrt {16 - 16{{\sin }^2}t} \cos tdt = 16\mathop \smallint \limits_0^{\frac{\pi }{4}} {\cos ^2}tdt = 8\mathop \smallint \limits_0^{\frac{\pi }{4}} \left( {1 + \cos 2t} \right)dt$

Đặt $x = 2\sin t \Rightarrow dx = 2\cos tdt$

Đổi cận:$\left\{ {\begin{array}{*{20}{c}}{x = 0 \Rightarrow t = 0}\\{x = 1 \Rightarrow t = \frac{\pi }{6}}\end{array}} \right.$

Khi đó ta có: $I = \mathop \smallint \limits_0^{\frac{\pi }{6}} \frac{{2\cos tdt}}{{\sqrt {4 - 4{{\sin }^2}t} }} = \mathop \smallint \limits_0^{\frac{\pi }{6}} \frac{{2\cos tdt}}{{2\cos t}} = \mathop \smallint \limits_0^{\frac{\pi }{6}} dt$

Đặt $t = {e^x} \Rightarrow dt = {e^x}dx$

Đổi cận:$\left\{ {\begin{array}{*{20}{c}}{x = - 1 \Rightarrow t = {e^{ - 1}}}\\{x = 2 \Rightarrow t = {e^2}}\end{array}} \right.$

Khi đó

$\begin{array}{l}I = \int\limits_{{e^{ - 1}}}^{{e^2}} {\frac{{dt}}{{t + 2}}} = ln|t + 2|\left| {_{{e^{ - 1}}}^{{e^2}}} \right. = ln({e^2} + 2) - ln({e^{ - 1}} + 2) = ln\frac{{{e^2} + 2}}{{{e^{ - 1}} + 2}}\\ = \ln \frac{{{e^2} + 2}}{{\frac{1}{e} + 2}} = \ln \frac{{2e + {e^3}}}{{2e + 1}} = \ln \frac{{ae + {e^3}}}{{ae + b}}\end{array}$

$\Rightarrow \left\{ {\begin{array}{*{20}{c}}{ae + {e^3} = 2e + {e^3}}\\{ae + b = 2e + 1}\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{a = 2}\\{b = 1}\end{array}} \right.$

Ta có$3\sin x + \cos x = A\left( {2\sin x + 3\cos x} \right) + B\left( {2\cos x - 3\sin x} \right)$

$\Leftrightarrow 3sinx + cosx = (2A - 3B)sinx + (3A + 2B)cosx$

$\Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{2A - 3B = 3}\\{3A + 2B = 1}\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{A = \frac{9}{{13}}}\\{B = - \frac{7}{{13}}}\end{array}} \right.$

Nên

$3\sin x + \cos x = \frac{9}{{13}}\left( {2\sin x + 3\cos x} \right) - \frac{7}{{13}}\left( {2\cos x - 3\sin x} \right)$

Từ đó ta có

$\int\limits_0^{\frac{\pi }{2}} {\frac{{3sinx + cosx}}{{2sinx + 3cosx}}} dx = \int\limits_0^{\frac{\pi }{2}} {\frac{{\frac{9}{{13}}(2sinx + 3cosx) - \frac{7}{{13}}(2cosx - 3sinx)}}{{2sinx + 3cosx}}dx}$

$\begin{array}{l} = \frac{9}{{13}}\int\limits_0^{\frac{\pi }{2}} {dx - \frac{7}{{13}}} \int\limits_0^{\frac{\pi }{2}} {\frac{{2cosx - 3sinx}}{{2sinx + 3cosx}}} dx\\ = \frac{{9\pi }}{{26}} - \frac{7}{{13}}\int\limits_0^{\frac{\pi }{2}} {\frac{1}{{2sinx + 3cosx}}} d(2sinx + 3cosx)\\ = \frac{{9\pi }}{{26}} - \frac{7}{{13}}ln|2sinx + 3cosx|\left| {_0^{\frac{\pi }{2}}} \right.\\ = \frac{{9\pi }}{{26}} - \frac{7}{{13}}ln2 + \frac{7}{{13}}ln3\end{array}$

Suy ra$b = \frac{7}{{13}};c = \frac{9}{{26}} \Rightarrow \frac{b}{c} = \frac{{14}}{9}$

Đặt$t = \sin 2x \Rightarrow dt = 2\cos 2xdx \Rightarrow - \frac{1}{2}dt = \left( {2{{\sin }^2}x - 1} \right)dx$

Đổi cận: $\left\{ {\begin{array}{*{20}{c}}{x = 0 \Rightarrow t = 0}\\{x = \frac{\pi }{4} \Rightarrow t = 1}\end{array}} \right.$

$\Rightarrow I = \mathop \smallint \limits_0^{\frac{\pi }{4}} \left( {2{{\sin }^2}x - 1} \right)f\left( {\sin 2x} \right)dx = - \frac{1}{2}\mathop \smallint \limits_0^1 f\left( t \right)dt = - \frac{1}{2}.1 = - \frac{1}{2}.$

Ta có:$I = \mathop \smallint \limits_0^\pi xf\left( {\sin x} \right)dx = \mathop \smallint \limits_0^{\frac{\pi }{2}} xf\left( {\sin x} \right)dx + \mathop \smallint \limits_{\frac{\pi }{2}}^\pi xf\left( {\sin x} \right)dx$

Xét ${I_1} = \mathop \smallint \limits_{\frac{\pi }{2}}^\pi xf\left( {\sin x} \right)dx$đặt$t = \pi - x \Rightarrow dt = - dx$

Đổi cận: $\left\{ {\begin{array}{*{20}{c}}{x = \frac{\pi }{2} \Rightarrow t = \frac{\pi }{2}}\\{x = \pi \Rightarrow t = 0}\end{array}} \right.$

Khi đó ta có:

$\begin{array}{*{20}{l}}{{I_1} = - \mathop \smallint \limits_{\frac{\pi }{2}}^0 \left( {\pi - t} \right)f\left( {\sin \left( {\pi - t} \right)} \right)\,dt}\\{\,\,\,\,\,\, = \mathop \smallint \limits_0^{\frac{\pi }{2}} \left( {\pi - t} \right)f\left( {\sin t} \right)\,dt}\\{\,\,\,\,\,\, = \mathop \smallint \limits_0^{\frac{\pi }{2}} \left( {\pi - x} \right)f\left( {\sin x} \right)\,dx}\\{\,\,\,\,\,\, = \pi \mathop \smallint \limits_0^{\frac{\pi }{2}} f\left( {\sin x} \right)\,dx - \mathop \smallint \limits_0^{\frac{\pi }{2}} xf\left( {\sin x} \right)\,dx}\end{array}$

$\begin{array}{*{20}{l}}{ \Rightarrow I = \mathop \smallint \limits_0^{\frac{\pi }{2}} xf\left( {\sin x} \right)dx + \pi \mathop \smallint \limits_0^{\frac{\pi }{2}} f\left( {\sin x} \right)\,dx - \mathop \smallint \limits_0^{\frac{\pi }{2}} xf\left( {\sin x} \right)\,dx}\\{ \Rightarrow I = \pi \mathop \smallint \limits_0^{\frac{\pi }{2}} f\left( {\sin x} \right)\,dx = 5\pi .}\end{array}$

$I = \mathop \smallint \limits_0^2 f\left( x \right).f'\left( x \right)dx$

Đặt$f\left( x \right) = t \Rightarrow dt = f'\left( x \right)dx$

Đổi cận:$\left\{ {\begin{array}{*{20}{c}}{x = 0 \Rightarrow t = \sqrt 5 }\\{x = 2 \Rightarrow t = \sqrt {11} }\end{array}} \right.$

$\Rightarrow I = \int\limits_{\sqrt 5 }^{\sqrt {11} } {tdt = \frac{{{t^2}}}{2}} \left| {_{\sqrt 5 }^{\sqrt {11} }} \right. = \frac{1}{2}(11 - 5) = 3.$

Ta có$\mathop \smallint \limits_{ - 1}^1 f\left( {\left| {2x - 1} \right|} \right)dx = \mathop \smallint \limits_{ - 1}^{\frac{1}{2}} f\left( {1 - 2x} \right)dx + \mathop \smallint \limits_{\frac{1}{2}}^1 f\left( {2x - 1} \right)dx$

$\Rightarrow I = - \frac{1}{2}\mathop \smallint \limits_{ - 1}^{\frac{1}{2}} f\left( {1 - 2x} \right)d\left( {1 - 2x} \right) + \frac{1}{2}\mathop \smallint \limits_{\frac{1}{2}}^1 f\left( {2x - 1} \right)d\left( {2x - 1} \right)$

$\begin{array}{*{20}{l}}{ \Leftrightarrow I = - \frac{1}{2}\mathop \smallint \limits_3^0 f\left( t \right)dt + \frac{1}{2}\mathop \smallint \limits_0^1 f\left( t \right)dt}\\{ \Leftrightarrow I = \frac{1}{2}\mathop \smallint \limits_0^3 f\left( t \right)dt + \frac{1}{2}\mathop \smallint \limits_0^1 f\left( t \right)dt = \frac{1}{2}\left( {2 + 6} \right) = 4}\end{array}$