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We have to find an integral that can be solved by using integration by parts with $u=x$ and another integral that can be solved by using integration by parts with$dv=xdx$

• If $u$and $v$are differentiable functions, the formula for integration by parts is $鈭�udv=uv-鈭�vdu$
• In the given integral, the values of $u$and $dv$are chosen such that the integration process becomes simple.
• While selecting the function for $u$, the following order of preference can be followed:

- Logarithmic function

- Inverse trigonometric

- Algebraic function

- Trigonometric

- Exponential function

In order to have $u=x$, the integrand can have function such as trigonometric and exponential functions along with x.

One of the examples of an integral that can be solved by integration by parts with$u=x$is$鈭�x{e}^{x}dx$

Here, $dv=xdx$which is an algebraic function.

Choosing $u=\ln x$gives $du=\frac{1}{x}dx$. This simplifies the integration process.

Thus, the required integral is$鈭�x\ln xdx$

• The integral that can be solved by integration by parts with $u=x$is $鈭�x{e}^{x}dx$
• The integral that can be solved by integration by parts with $dv=xdx$is$鈭�x\ln xdx$