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Worked solution

Evaluate exact definite integrals

Example 3 (Self-Reading). Without the use of a calculator, evaluate (a) ∫023(1+x)2 dx\int_0^2 3(1+x)^2\,dx, (b) ∫0π/2cos⁡x dx\int_0^{\pi/2}\cos x\,dx, (c) ∫12e1+x dx\int_1^2e^{1+x}\,dx.

CJC · Lecture · Example 3

(a)

∫023(1+x)2 dx=[(1+x)3]02=33−13=26\int_0^23(1+x)^2\,dx=[(1+x)^3]_0^2=3^3-1^3=26.

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(b)

∫0π/2cos⁡x dx=[sin⁡x]0π/2=sin⁡(π/2)−sin⁡0=1\int_0^{\pi/2}\cos x\,dx=[\sin x]_0^{\pi/2}=\sin(\pi/2)-\sin0=1.

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(c)

∫12e1+x dx=[e1+x]12=e3−e2=e2(e−1)\int_1^2e^{1+x}\,dx=[e^{1+x}]_1^2=e^3-e^2=e^2(e-1).

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Common mistakes

Apply the continuous-interval theorem only when its conditions hold; do not substitute across a singularity.

Read the hint