# Integration of Cosx

Here you will learn proof of integration of cosx or cos x and examples based on it.

Let’s begin –

## Integration of Cosx or Cos x

The integration of cosx is sinx + C

where C is the integration constant.

i.e. $$\int$$ (cosx) dx = sin x + C

Proof :

We will prove this formula using differentiation,

Let $$d\over dx$$(sin x + C) = $$d\over dx$$ sin x + $$d\over dx$$ C

Using differentiation formula,

$$d\over dx$$ sin x = cos x and differentation of constant is 0.

$$\implies$$ $$d\over dx$$(sin x + C) = $$d\over dx$$ sin x + $$d\over dx$$ C

$$\implies$$ $$d\over dx$$(sin x + C) = cos x + 0

We can also write it as,

cos x = $$d\over dx$$(sin x + C)

Now, integrating on both sides,

$$\int$$ cos x = $$\int$$ $$d\over dx$$(sin x + C)

We know that integration and differentiation both are reciprocals of each other, so in right hand side expression they cancel each other and we get,

Hence, $$\int$$ cos x = sin x + C

Example : Prove that $$\int$$ cos (ax + b) = $$1\over a$$ sin(ax + b) + C.

Solution : We have,

I = $$\int$$ cos (ax + b) dx

Let ax + b = t, Then , d(ax + b) = dt $$\implies$$ adx = dt

$$\implies$$ dx = $$1\over a$$ dt

Putting ax + b = t and  dx = $$1\over a$$ dt , we get

$$\int$$ cos (ax + b) dx = $$1\over a$$ $$\int$$ cos t dt

= $$1\over a$$ sin t + C

= $$1\over a$$ sin(ax + b) + C

Hence, $$\int$$ cos (ax + b) = $$1\over a$$ sin(ax + b) + C

### Related Questions

What is the Differentiation of cos x ?

What is the Integration of Cos Inverse x ?

What is the Differentiation of cos inverse x ?