AI HL · AHL 3
0 of 78 skills done
Definite integrals by hand and with the calculator
AHL 3.4
Screenshot and video to come
Screenshot and video to come
Evaluate \(\int_{-3}^{2} x^2\,dx\) without a calculator.
Have a go first. Then open the working one line at a time.
\(\int_{-3}^{2}x^2\,dx = \left[\frac{x^3}{3}\right]_{-3}^{2}\)
No + c is needed in a definite integral: it cancels. Write the limits on the bracket.
\(= \frac{2^3}{3} - \frac{(-3)^3}{3}\)
Upper limit first, then subtract the lower limit. Brackets round −3.
\(= \frac{8}{3} - (-9) = \frac{35}{3}\)
Subtracting a negative adds. The answer is 11.7 to 3 significant figures.
Evaluate \(\int_{1}^{27} x^{-1/3}\,dx\) without a calculator.
Have a go first. Then open the working one line at a time.
\(-\frac13 + 1 = \frac23\)
The new power is 2/3.
\(\int_{1}^{27} x^{-1/3}\,dx = \left[\frac{x^{2/3}}{2/3}\right]_1^{27} = \left[\frac32 x^{2/3}\right]_1^{27}\)
Dividing by 2/3 is multiplying by 3/2.
\(= \frac32(27)^{2/3} - \frac32(1)^{2/3}\)
\(27^{2/3}\) is the cube root of 27, squared: \(3^2=9\).
\(= \frac32\times 9 - \frac32\times 1 = 12\)
A whole number: the limits were chosen to be perfect cubes.
Find the exact value of \(\int_{\pi/4}^{\pi} \sin x\,dx\).
Have a go first. Then open the working one line at a time.
\(\int_{\pi/4}^{\pi}\sin x\,dx = \left[-\cos x\right]_{\pi/4}^{\pi}\)
Radians: the limits are in terms of π.
\(= (-\cos\pi) - \left(-\cos\frac{\pi}{4}\right)\)
Upper limit minus lower limit.
\(= 1 + \frac{\sqrt2}{2}\)
cos π = −1 and cos(π/4) = √2/2. Exact value: 1 + √2/2 ≈ 1.71.
Find the exact value of \(\int_{\ln 3}^{\ln 8} 2e^x\,dx\).
Have a go first. Then open the working one line at a time.
\(\int_{\ln 3}^{\ln 8}2e^x\,dx = \left[2e^x\right]_{\ln 3}^{\ln 8}\)
The integral of \(e^x\) is \(e^x\).
\(= 2e^{\ln 8} - 2e^{\ln 3}\)
\(e\) and \(\ln\) undo each other: \(e^{\ln k}=k\).
\(= 2\times 8 - 2\times 3 = 10\)
An exact whole number.
Find the exact value of \(\int_{3}^{12} \frac{5}{x}\,dx\), then give it to 3 significant figures.
Have a go first. Then open the working one line at a time.
\(\int_{3}^{12}\frac5x\,dx = \left[5\ln x\right]_3^{12}\)
The integral of 1/x is ln x (x is positive here).
\(= 5\ln 12 - 5\ln 3\)
Upper limit minus lower limit.
\(= 5\ln\frac{12}{3} = 5\ln 4\)
ln a − ln b = ln(a/b). Exact value 5 ln 4 ≈ 6.93.
Evaluate \(\int_{0}^{2} e^{-x^2/2}\,dx\), giving your answer to 3 significant figures.
Have a go first. Then open the working one line at a time.
\(e^{-x^2/2}\) has no antiderivative we can write
None of the standard integrals gives it, so do not try by hand.
From the calculator: \(\int_0^2 e^{-x^2/2}\,dx = 1.196288\)
Put the whole exponent in brackets: \(e^{(-x^2/2)}\).
\(\approx 1.20\)
3 significant figures.
Find the exact value of \(\int_0^{\pi/6}\frac{8}{\cos^2(2x)}\,dx\), then give it to 3 significant figures.
Have a go first. Then open the working one line at a time.
\(\int\frac{8}{\cos^2(2x)}\,dx=\frac82\tan(2x)=4\tan(2x)\)
Divide by 2, the coefficient of x.
\(\left[4\tan 2x\right]_0^{\pi/6}=4\tan\frac\pi3-4\tan 0\)
Upper limit minus lower limit.
\(\tan\frac\pi3=\sqrt3,\quad 4\sqrt3-0=4\sqrt3\)
Use the exact value of tan π/3.
\(4\sqrt3=6.93\)
Check on the calculator in radians.