Solution assignment 10 Integration by parts

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Assignment 10

In assignment 3 it was asked to calculate the following integral and we applied integration by parts. Is there another way and if so, which?

\displaystyle\int{\sin^2(x)}dx

Solution

An important formula from trigonometry is:

\cos(2x)=1-2\sin^2(x)

Or, written differently:

\sin^2(x)=\displaystyle\frac{1}{2}(1-\cos(2x))

This integral can be calculated using standard functions and substitution:

\displaystyle\int{\sin^2(x)}dx=

=\displaystyle\frac{1}{2}\displaystyle\int{(1-\cos(2x))}dx=

=\displaystyle\frac{1}{2}(x-\displaystyle\frac{1}{2}\sin(2x))+C

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