Question

What is the integration of $\sin wt$?

Answer 1

Find the integration of $\sin wt$.

$$\begin{gathered} \mathrm{Let},wt=u \\ \Rightarrow wdt=du\left[\because Differenciationw.r.tox\right] \end{gathered}$$
Put the above values in $\int \left(\sin wt\right)dt$ ,to find the integration of $\sin wt$:

$\begin{array}{rcl}\int \left(\sin wt\right)dt& =& \frac{1}{w}\int \sin udu\\ & \Rightarrow & \int \left(\sin wt\right)dt=\frac{1}{w}\left(-\cos u\right)+C\left[\because \int sinu=-cosu,Cisaconstant\right]\\ & \Rightarrow & \int \left(\sin wt\right)dt=\frac{1}{w}\left[-\cos \left(wt\right)\right]+C\left[\because u=wt\right]\end{array}$

Hence, the required integration is $\begin{array}{rcl}\int \left(\sin wt\right)dt& =& \frac{1}{w}\left[-\cos \left(wt\right)\right]+C\end{array}$ where $C$ is a constant.