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Let $m,n$ be coprime positive integers. Show that $\varphi(nm)=\varphi(n)\varphi(m)$. Use this and parts (a), (b) to find a formula for $\varphi(n)$ for an arbitrary $n\geq 2$.

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A and T factors

Compute the integral: $\displaystyle\int x\cos x\,dx$

Solution

(1) Choose $u=x$.

Set $u(x)=x$ because $x$ simplifies when differentiated. (By the trick: $x$ is Algebraic, i.e. more “$u$”, and $\cos x$ is Trig, more “$v$”.)

Remaining factor must be $v'$:

$$v'(x)=\cos x$$

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(2) Compute $u'$ and $v$.

Derive $u$:

$$u'=1$$

Antiderive $v'$:

$$v=\sin x$$

Obtain chart:

$$\begin{array}{c|c}u=x&v'=\cos x\\\hline u'=1&v=\sin x\end{array} \begin{array}{l}\quad\longrightarrow\quad\int u\cdot v'\qquad\text{original} \\ \quad\longrightarrow\quad\int u'\cdot v\qquad\text{final} \end{array}$$

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(3) Plug into IBP formula.

Plug in all data:

$$\int x\cos x\,dx=x\sin x-\int 1\cdot\sin x\,dx$$

Compute integral on RHS:

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

Note: the point of IBP is that this integral is easier than the first one!

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(4) Final answer is: $\;x\sin x+\cos x + C$

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