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Worked solution

Approximate an integral with a square root

Evaluate ∫0422+x dx\int_0^4\frac2{2+\sqrt x}\,dx.

CJC · Lecture · Self-Practice 2

Calculator result

Enter lower limit 00, upper limit 44 and integrand 2/(2+x)2/(2+\sqrt x) in the calculator integral template. The value is approximately 2.4548225562.454822556, hence 2.452.45 to three significant figures, as printed.

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Optional exact check

Independently, put t=xt=\sqrt x, so dx=2t dtdx=2t\,dt and the limits are 0,20,2. Then ∫024t2+t dt=∫02(4−8t+2) dt=[4t−8ln⁡(t+2)]02=8−8ln⁡2\int_0^2\frac{4t}{2+t}\,dt=\int_0^2(4-\frac8{t+2})\,dt=[4t-8\ln(t+2)]_0^2=8-8\ln2, agreeing with the printed decimal.

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Common mistakes

Keep the entire denominator 2+sqrt(x) inside parentheses when entering the expression.

Read the hint