91Ó°ÊÓ

Here, we are asked to find the integral which can be solved by integration by parts by taking $u=\sin x$and another integral which can be solved by taking$dv=\sin xdx$

If $u$and $v$are differentiable functions, then the formula for integration by parts is $∫udv=uv-∫vdu$

The selection of $u$and $dv$must be done in such a way that, the integration becomes simpler during the solution and not complicated.

While choosing the function $u$, the following order of preference can be kept in mind:

• Inverse trigonometric
• Logarithmic
• Algebraic
• Trigonometric
• Exponential

According to the order of preference of the first function $u$, when the integrand has a product of $e^x$and $\sin x$, choosing $u=\sin x$, makes the process of integration by parts easier.

Therefore, the integral that can be solved by integration by parts by taking $u=\sin x$is$∫e^x\sin xdx$

In order to choose $dv=\sin xdx$, the term $u$can be inverse trigonometric function, logarithmic, algebraic, trigonometric or exponential function.

Therefore, one of the example for an integral with $dv=\sin xdx$that can be solved by integration by parts is $∫x^2\sin xdx$

• The integral that can be solved by integration by parts with $u=\sin x$is $∫e^x\sin xdx$
• The integral that can be solved by integration by parts with$dv=\sin xdx$$∫x^2\sin xdx$