$\dfrac{\csc x}{\log\left( an \frac{x}{2} ight)}$

Question

$\dfrac{\csc x}{\log\left(	an \frac{x}{2}ight)}$

Accepted Answer

> [!TLDR]
>
> A short computation shows the derivative of $\log(	an(x/2))$ is precisely $\csc x$, so the numerator is exactly $du$.

**Step 1. Differentiate the candidate $u$.** $\dfrac{d}{dx}\log\left(	an\dfrac x2ight)=\dfrac1{	an(x/2)}\cdot\sec^2\dfrac x2\cdot\dfrac12=\dfrac1{2\sin(x/2)\cos(x/2)}=\dfrac1{\sin x}=\csc x$ (using $\sin x=2\sin\frac x2\cos\frac x2$).

**Step 2. Substitute.** Let $u=\log\left(	an\dfrac x2ight)$, so $du=\csc x\,dx$, exactly the numerator.

**Step 3. Integrate.** $\displaystyle\int\dfrac{du}{u}=\log|u|+c$. …

Q. $\dfrac{\csc x}{\log\left(\tan \frac{x}{2}\right)}$

Concept understanding — Integration by Substitution

Step 1 — Use the standard form for a linear substitution

Step 2 — Identify the constants

Step 3 — Substitute

Step 1 — Prepare for substitution

Step 2 — Substitute u = x⁵

Step 3 — Partial fractions and integrate

Q.csc⁡xlog⁡(tan⁡x2)\dfrac{\csc x}{\log\left(\tan \frac{x}{2}\right)}log(tan2x​)cscx​

🎓Unlock everything free for 14 days

Q. $\dfrac{\csc x}{\log\left(\tan \frac{x}{2}\right)}$

Unlock everything free for 14 days