Question

What is `int cos(7x+pi)-sin(5x-pi)`?

Answer 1

`-(sin7x)/7-(cos5x)/5+C`

Explanation:

Before calculating the integral let us simplify the trigonometric expression using some trigonometric properties we have:

Applying the property of `cos` that says:
`cos(pi+alpha)=-cosalpha`

`cos(7x+pi)=cos(pi+7x)`
So,
`color(blue)(cos(7x+pi)=-cos7x)`

Applying two properties of `sin` that says:
`sin(-alpha)=-sinalpha`and
`sin(pi-alpha)=sinalpha`

We have:
`sin(5x-pi)=sin(-(pi-5x))=-sin(pi-5x)` since
`sin(-alpha)=-sinalpha`
`-sin(pi-5x)=-sin5x`
Since`sin(pi-alpha)=sinalpha`
Therefore,
`color(blue)(sin(5x-pi)=-sin5x)`

First Substitute the simplified answers then compute the integral:

`color(red)(intcos(7x+pi)-sin(5x-pi)`
`=int-cos(7x)-(-sin5x)`
`=int-cos7x+sin5x`
`=-intcos7x+intsin5x`
`color(red)(=-(sin7x)/7-(cos5x)/5+C` ( where `C`is a constant number).