The integral of cos root x dx is denoted by ∫cos(√x)dx, and it is equal to ∫cos(√x)dx = 2[√xsin(√x)+ cos(√x)]+C where C is an integration constant. Here we will learn how to integrate cos root x.
The integral formula of cos root x is given below.
$\int \cos \sqrt{x} dx$ $= 2[\sqrt{x} \sin \sqrt{x}+\cos \sqrt{x}]+C$ |
Integration of cos root x
Question: Find the integral ∫cos(√x)dx.
Answer:
Let us put
I = ∫cos(√x)dx.
Put √x=t, that is, x=t2.
Differentiating both sides, we get
dx = 2tdt
So, I = ∫cos(√x)dx = 2 ∫tcos(t) dt
To compute this integral, we use the integration by parts formula: ∫uv dx = u ∫v dx – ∫[$\frac{du}{dx}$∫v dx] dx (where u, v are functions of x).
Take
u= t, v=cost.
Thus,
I = 2 ∫tcos(t) dt
= 2 [t ∫cost dt – ∫{$\frac{dt}{dt}$∫cost dt} dt ] + C
= 2[t sint – ∫sint dt] + C
= 2[t sint + cost] + C
= 2[√x sin(√x) + cos(√x)] + C.
So the integral of cos(root x) is equal to ∫cos(√x)dx = 2[√x sin(√x) + cos(√x)] + C where C is a constant.
More Integrals:
Integration of root tanx + root cotx
Integral of square root of tanx
FAQs
Q1: What is the integration of cos root x?
Answer: The integration of cos root x is given by ∫cos(√x)dx = 2[√xsin(√x)+ cos(√x)]+C where C is a constant.