Differentiate $f(x)=\int_{\sqrt{x}}^{\arcsin x} \ln\theta d \theta$
Answer
You need to use the Leibniz rule:
\[(\int_{g(x)}^{h(x)}f(\theta)d\theta)'=h'(x)f(h(x))-g'(x)f(g(x)).\]
Using this formula we get
\[f'(x)=(\arcsin x)' \ln(\arcsin x)-(x^{1/2})'\ln(x^{1/2})\]
\[=\frac{1}{\sqrt{1-x^2}} \ln(\arcsin x)-\frac{1}{2}x^{-1/2}\ln (x^{1/2}).\]

4.8K
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 1776 views
- $2.00
Related Questions
- Stuck on this and need the answer for this problem at 6. Thanks
- Help with Norms
- Find $\int x \sqrt{1-x}dx$
- Compounding interest of principal P, where a compounding withdrawal amount W get withdrawn from P before each compounding of P.
- Beginner Question on Integral Calculus
- Let $R$ be an integral domain and $M$ a finitely generated $R$-module. Show that $rank(M/Tor(M))$=$rank(M)$
- Evaluate the limit Please explain all steps
- Minimizing the cost of building a box