Question

tính nguyên hàm (sd pp nguyên hàm từng phần)

1. \(\int\dfrac{x}{\sin^2x}dx\)

2. \(\int\dfrac{x+1}{e^x}dx\)

3. \(\int x.\sin x.\cos xdx\)

4. \(\int e^x\sin xdx\)

Answer 1

1. Đặt \(\left\{{}\begin{matrix}u=x\\dv=\dfrac{dx}{sin^2x}\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}du=dx\\v=-cotx\end{matrix}\right.\)

Do đó I= \(-x.cotx+\int cotxdx\)= \(-xcotx+ln\left|sinx\right|\)

Answer 2

2. Đặt \(\left\{{}\begin{matrix}u=x+1\\dv=\dfrac{dx}{e^x}\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}du=dx\\v=-e^{-x}\end{matrix}\right.\)

Do đó I= \(-\left(x+1\right)e^{-x}+\int e^{-x}dx\)=\(-\left(x+1\right)e^{-x}-e^{-x}\)

=\(-\left(x+2\right)e^{-x}\)

Answer 3

3. Đặt \(\left\{{}\begin{matrix}u=x\\dv=sinx.cosx.dx\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}du=dx\\dv=\dfrac{1}{4}sin2x.d\left(2x\right)\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}du=dx\\v=\dfrac{-cos2x}{8}\end{matrix}\right.\), do đó I= \(\dfrac{-x.cos2x}{8}+\int\dfrac{cos2x}{8}dx\)

=\(\dfrac{-x.cos2x}{8}+\int\dfrac{cos2x}{16}d\left(2x\right)\)= \(\dfrac{-x.cos2x}{8}+\dfrac{sin2x}{32}\)

Answer 4

4. Đặt \(\left\{{}\begin{matrix}u=sinx\\dv=e^xdx\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}du=cosx.dx\\v=e^x\end{matrix}\right.\)

Do đó I = \(e^xsinx-\int e^xcosx.dx\)

đặt I' = \(\int e^xcosx.dx\) và \(\left\{{}\begin{matrix}a=cosx\\db=e^xdx\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}da=-sinx.dx\\b=e^x\end{matrix}\right.\)

suy ra I' = \(e^xcosx+\int e^xsinx.dx\)= \(e^xcosx+I\)

\(\Rightarrow I=e^xsinx-I'=e^xsinx-e^xcosx-I\)

\(\Rightarrow I=\dfrac{\left(sinx+cosx\right)e^x}{2}\)