ĐGNL ĐHQG Hà Nội - Tư duy định lượng - Sử dụng phương pháp tích phân từng phần để tính tích phân

Đặt\(\left\{ {\begin{array}{*{20}{c}}{u = f(x)}\\{dv = g\prime (x)dx}\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{du = f\prime (x)dx}\\{v = g(x)}\end{array}} \right.\) khi đó

\[I = f\left( x \right).g\left( x \right)\left| {_a^b} \right. - \int\limits_a^b {f'\left( x \right)} .g\left( x \right)dx\]

Đáp án cần chọn là: C

Đặt\(\left\{ {\begin{array}{*{20}{c}}{u = {x^2}}\\{dv = cosxdx}\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{du = 2xdx}\\{v = sinx}\end{array}} \right.\)khi đó\[I = {x^2}sinx\left| {_0^{\frac{\pi }{2}}} \right. - 2\int\limits_0^{\frac{\pi }{2}} {xsinxdx} \]

Đáp án cần chọn là: B

Đặt \(\left\{ {\begin{array}{*{20}{c}}{u = g(x)}\\{dv = f\prime (x)dx}\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{du = g\prime (x)dx}\\{v = f(x)}\end{array}} \right.\)

Khi đó

\[\int\limits_0^1 {g\left( x \right)} .f'\left( x \right)dx = \left[ {g(x).f(x)} \right]\left| {_0^1} \right. - \int\limits_0^1 {g'\left( x \right)} .f\left( x \right)dx\]

\( \Leftrightarrow \left[ {g(x).f(x)} \right]\left| {_0^1} \right. = 3\)

Mặt khác\(I = \int\limits_0^1 {\left[ {f\left( x \right).g\left( x \right)} \right]} 'dx = \left[ {f\left( x \right).g\left( x \right)} \right]\left| {_0^1} \right. \Rightarrow I = 3\)

Đáp án cần chọn là: C

Đáp án cần chọn là: A

Đặt\(\left\{ {\begin{array}{*{20}{c}}{u = x + lnx}\\{dv = \frac{{dx}}{{{{(x + 1)}^3}}}}\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{du = \frac{{x + 1}}{x}dx}\\{v = - \frac{1}{{2{{(x + 1)}^2}}}}\end{array}} \right.\)

Khi đó\[I = - \frac{{x + lnx}}{{2{{(x + 1)}^2}}}\left| {_1^2} \right. + \int\limits_1^2 {\frac{{x + 1}}{x}.\frac{1}{{2{{(x + 1)}^2}}}} dx\]

\[ = - \frac{{2 + \ln 2}}{{18}} + \frac{1}{8} + \frac{1}{2}\mathop \smallint \limits_1^2 \frac{{{\rm{d}}x}}{{x\left( {x + 1} \right)}}\]

\[ = - \frac{{2 + \ln 2}}{{18}} + \frac{1}{8} + \frac{1}{2}\mathop \smallint \limits_1^2 \left( {\frac{1}{x} - \frac{1}{{x + 1}}} \right){\rm{d}}x.\]

\( = - \frac{{2 + ln2}}{{18}} + \frac{1}{8} + \frac{1}{2}(ln|x| - ln|x + 1|)\left| {_1^2} \right.\)

\(\begin{array}{l} = \frac{1}{{72}} - \frac{1}{{18}}ln2 + \frac{1}{2}(ln2 - ln3 + ln2)\\ = \frac{1}{{72}} + \frac{{17}}{{18}}ln2 - \frac{1}{2}\ln 3\\ = a + b.ln2 - c.ln3\end{array}\)

Vậy\(\left\{ {\begin{array}{*{20}{c}}{a = \frac{1}{{72}}}\\{b = \frac{{17}}{{18}}}\\{c = \frac{1}{2}}\end{array}} \right. \Rightarrow \frac{c}{a} = \frac{1}{2}:\frac{1}{{72}} = 36\)

Đáp án cần chọn là: D

Ta có :\[{\left( {x\sin x + \cos x} \right)^\prime } = \sin x + x\cos x - \sin x = x\cos x\]

\[ \Rightarrow I = \mathop \smallint \limits_0^{\frac{\pi }{4}} \frac{{{x^2}}}{{{{\left( {x\sin x + \cos x} \right)}^2}}}{\rm{d}}x = \mathop \smallint \limits_0^{\frac{\pi }{4}} \frac{{\frac{x}{{\cos x}}.x\cos x}}{{{{\left( {x\sin x + \cos x} \right)}^2}}}dv\]

Đặt\(\left\{ {\begin{array}{*{20}{c}}{u = \frac{x}{{cosx}}}\\{dv = \frac{{xcosx}}{{{{(xsinx + cosx)}^2}}}dx}\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{du = \frac{{xsinx + cosx}}{{co{s^2}x}}dx}\\{v = \frac{1}{{xsinx + cosx}}}\end{array}} \right.\)

Khi đó

\(I = - \frac{x}{{cosx}}.\frac{1}{{xsinx + cosx}}\left| {_0^{\frac{\pi }{4}}} \right. + \int\limits_0^{\frac{\pi }{4}} {\frac{{dx}}{{co{s^2}x}}} \)

\( = \frac{{ - \frac{\pi }{4}}}{{\frac{{\sqrt 2 }}{2}}}.\frac{1}{{\frac{\pi }{4}\frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2}}} + tanx\left| {_0^{\frac{\pi }{4}}} \right.\)

\( = \frac{{ - \frac{\pi }{4}}}{{\frac{1}{2}\left( {\frac{\pi }{4} + 1} \right)}} + 1 = \frac{{ - 2\pi }}{{\left( {\pi + 4} \right)}} + 1 = \frac{{4 - \pi }}{{4 + \pi }} \Rightarrow m = 4\)

Đáp án cần chọn là: C

Đặt

\(\left\{ {\begin{array}{*{20}{c}}{u = ln(3sinx + cosx)}\\{dv = \frac{{dx}}{{si{n^2}x}}}\end{array}} \right.\)

\( \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{du = \frac{{3cosx - sinx}}{{3sinx + cosx}}dx}\\{v = - cotx - 3 = - \frac{{3sinx + cosx}}{{sinx}}}\end{array}} \right.\)

Khi đó

\[I = [ - (cotx + 3)ln(3sinx + cosx)]\left| {_{\frac{\pi }{4}}^{\frac{\pi }{2}}} \right. + \int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {\frac{{3cosx - sinx}}{{sinx}}} dx\]

\[ = 4.ln2\sqrt 2 - 3.ln3 - \int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {dx + 3\int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {\frac{{d(sinx)}}{{sinx}}} .} \]

\[ = 4.ln2\sqrt 2 - 3.ln3 - \int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {dx + 3ln|sinx|\left| {_{\frac{\pi }{4}}^{\frac{\pi }{2}}} \right.} \]

\[\begin{array}{l} = 4.ln2\sqrt 2 - 3.ln3 - \frac{\pi }{4} - 3.ln\frac{1}{{\sqrt 2 }}\\ = 12ln\sqrt 2 - 3ln3 - \frac{\pi }{4} + 3ln\sqrt 2 = 15.ln\sqrt 2 - 3.ln3 - \frac{\pi }{4}\end{array}\]

\( \Rightarrow \left\{ {\begin{array}{*{20}{c}}{m = 15}\\{n = - 3}\end{array}} \right. \Rightarrow m + n = 12\)

Đáp án cần chọn là: A

Đáp án cần chọn là: D