I've been given a problem by my math lecturer to think about:
Use substitution to find the exact value of: $\int_{0}^{\frac{1}{\sqrt{10}}}\frac{x}{\sqrt{1-25x^4}}dx$
How do you figure out what the value of $u$ should be for the substitution? Ive tried $u=5x$ but this doesn't work.
Is there a way I can generally work out what $u$ should be for any given integral?
2026-03-28 06:23:31.1774679011
What to substitute in any given integral?
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Try $u=5x^2$ so $\frac{du}{dx}=10x$. Then the integral becomes $$\frac1{10}\int_0^{1/2}\frac 1{\sqrt{1-u^2}}\,du$$ which should be easy to solve. It should help noticing that integrands of the form $\frac?{\sqrt{1-x^2}}$ are usually easy to solve, and that $25x^4=(5x^2)^2$.