# How do you integrate $(1 + \cos(x))^{5/2} dx$? [closed]

I tried substituting $$\cos(x) = 1 - 2\sin^2(x/2)$$ but still can't figure it out. Is there any other identity to help with this integration?

## closed as off-topic by Eevee Trainer, YuiTo Cheng, Brevan Ellefsen, Jean-Claude Arbaut, ShaileshMay 14 at 6:11

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• $1+\cos x=2\cos^2\frac{x}{2}$. – Nosrati May 8 at 6:54

## 1 Answer

You can use $$\cos(x)=2\cos^2(\frac{x}{2})-1$$ the next step can be to use $$\cos^5(\frac{x}{2})=\cos(\frac{x}{2})(1-\sin^4(\frac{x}{2}))$$

• Just remember about the absolute value: $\big(\cos^2\frac{x}{2}\big)^\frac52 = |\cos \frac{x}{2}|^5$ – Adam Latosiński May 8 at 7:08
• Okay. Just a query does it matter in an indefinite integral. – Archis Welankar May 8 at 7:58
• Yes it does; If you want to write the formula for the indefinite integral on the domain $D=\mathbb{R}$, that means that for some $x$ its derivative is supposed to be $(\cos \frac{x}{2})^5$ and for some other it is supposed to be $-(\cos \frac{x}{2})^5$. If you neglect the absolute value, you won't get that result. – Adam Latosiński May 8 at 8:38
• Thanks for answering. – Archis Welankar May 8 at 13:06