Bài tập 4 trang 126 SGK Giải tích 12

Bài tập 4 trang 126 SGK Giải tích 12

Câu a:

Đặt \(\left\{\begin{matrix} u=2-x\\ dv=sinx dx \end{matrix}\right.\Rightarrow \left\{\begin{matrix} du=-dx\\ v=-cosx \end{matrix}\right.\)

Áp dụng cộng thức tính nguyên hàm từng phần ta có:
\(\int (2-x)sinx dx=-(2-x)cosx-\int cosx dx\)

\(=(x-2)cosx-sinx+C\)

Câu b:

\(\int \frac{(x+1)^2}{\sqrt{x}}=\frac{x^2+2x+1}{x^{\frac{1}{2}}}dx\)

\(=\int \left ( x^\frac{3}{2}+2x^{\frac{1}{2}}+x^{-\frac{1}{2}} \right )dx\)

\(=\int x^{\frac{3}{2}}dx+2\int x^{\frac{1}{2}}dx+\int x^{-\frac{1}{2}}dx\)

\(=\frac{1}{1+\frac{3}{2}}x^{\frac{3}{2}+1}+\frac{2}{1 +\frac{1}{2}}x^{\frac{1}{2}+1}+\frac{1}{1-\frac{1}{2}}x^{-\frac{1}{2}+1}+C\)

\(=\frac{2}{5}x^\frac{5}{2}+ \frac{4}{3}x^{\frac{3}{2}}+2x^{\frac{1}{2}}+C.\)

Câu c:

\(\int \frac{e^{3x}+1}{e^x+1}dx=\int (e^{2x}-e^x+1)dx=\int e^{2x}dx- \int e^x dx +\int dx\)

\(=\frac{1}{2}\int e^{2x}d(2x)-\int e^xdx+\int dx=\frac{1}{2}e^{2x}-e^x+x+C\)

Câu d:

 \(\int \frac{1}{(sinx+cosx)^2}dx=\frac{1}{2}\int \frac{dx}{cos^2(x-\frac{\pi }{4})} =\frac{1}{2} tan \left ( x-\frac{\pi }{4} \right )+C\)

Câu e:
\(\int \frac{1}{\sqrt{1+x}+\sqrt{x}}dx=\int \frac{\sqrt{1+x}-\sqrt{x}}{1+x-x}dx\)

\(=\int (\sqrt{1+x}-\sqrt{x})dx=\int \sqrt{1+x}dx-\int \sqrt{x}dx\)

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

\(=\frac{2}{3}(1+x)^\frac{3}{2}-\frac{2}{3}x^{\frac{3}{2}}+C\)

Câu f:

\(\int \frac{1}{(1+x)(2-x)}dx=\frac{1}{3}\int \left ( \frac{1}{1+x}+ \frac{1}{2-x} \right )dx\)

\(=\frac{1}{3}\left ( \int \frac{dx}{1+x}+ \int \frac{dx}{2-x} \right )= \frac{1}{3}\left ( \int \frac{d(1+x)}{1+x} - \int \frac{d(2-x)}{2-x}\right )\)

\(=\frac{1}{3}\left ( ln \left | 1+x \right | - ln\left | 2-x \right | \right )+C =\frac{1}{3}ln \left | \frac{1+x}{2-x} \right |+C\)

-- Mod Toán 12