Change the limits
In \(\int_0^{\pi/2}\sin^2x\cos x\,dx\) put \(u=\sin x\): limits become 0 and 1, giving \(\int_0^1u^2du=\frac13\).
Integrals
Definite Integrals and the Fundamental Theorem
Understand
If \(F\) is any antiderivative of a continuous function \(f\) on \([a,b]\), the Fundamental Theorem of Calculus gives
\[\int_a^bf(x)\,dx=\Big[F(x)\Big]_a^b=F(b)-F(a).\]
The constant \(C\) cancels, so it is omitted. Example: \(\int_1^2(3x^2+1)\,dx=[x^3+x]_1^2=10-2=8\).
Substitution in definite integrals: change the limits along with the variable. \(\int_0^1\frac{2x}{1+x^2}dx\): with \(u=1+x^2\), limits \(1\to2\): \([\log u]_1^2=\log2\).
Key Concepts
- ∫ₐᵇ f = F(b) − F(a).
- Change limits when you substitute.
- Definite integral of a rate = total change.
Key Points
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Find an antiderivative, evaluate at the upper and lower limits, subtract.
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