Integration

Integration questions are the easy half of calculus: a standard integral, a definite integral, an area or an integration by substitution or parts. Learn the table and the three rules, and be sure to add +C for indefinite integrals.

Integrate term by term; constants pull out. \int(ax+b)^n dx=\frac{(ax+b)^{n+1}}{a(n+1)}+C and \int\frac{dx}{ax+b}=\frac{1}{a}\ln|ax+b|+C.

13.2 Methods: substitution and integration by parts

• Substitution: put u=g(x). In particular \int\frac{f'(x)}{f(x)}dx=\ln|f(x)|+C, so \int\frac{2x}{x^2+1}dx=\ln(x^2+1)+C.
• By parts: \int u\,dv=uv-\int v\,du, choosing u in the order Inverse trig, Log, Algebraic, Trig, Exponential (ILATE). Example: \int xe^x dx=(x-1)e^x+C; \int_1^e\ln x\,dx=1.

13.3 Definite integrals and area

• \int_a^bf\,dx=F(b)-F(a). Reversing limits changes the sign; \int_a^bf=\int_a^cf+\int_c^bf.
• Over [-a,a]: an odd function gives 0 (\int_{-1}^1x^3dx=0); an even function gives 2\int_0^a.
• \int_0^\pi\sin x\,dx=2, \int_0^{\pi/2}\cos x\,dx=1.
• Area between y=f(x), the x-axis and x=a,b is \int_a^b|f|dx; between two curves, \int_a^b(f-g)dx with f\ge g.

(a) Archimedes: slicing to find area

(b) Riemann: the integral as a limit

(c) Kaligandaki dam: force by integration

13.4 Eight traps in integration

Integration questions are checked by differentiating, so most errors are about a missing factor or a forgotten limit. These eight traps are the ones that cost marks in the model paper.

1. An indefinite integral needs +C; a definite integral does not. \int(3x^2+2x+1)\,dx=x^3+x^2+x+C, but \int_0^2(3x^2+2x+1)\,dx=8+4+2=14 with no constant. Examiners often put an option with and without C. Always check by differentiating your answer.
2. The power rule fails at n=-1. \int x^n\,dx=\frac{x^{n+1}}{n+1} for n\ne-1, and \int\frac1x\,dx=\ln|x|+C (note the absolute value). The integral of a constant k is kx. Do not forget to add 1 to the power before dividing.
3. Divide by the inner derivative. \int\cos2x\,dx=\frac12\sin2x+C and \int e^{ax}dx=\frac{e^{ax}}{a}+C. If you leave out the factor \frac1a your derivative check gives 2\cos2x, not \cos2x. This is the reverse of the chain rule.
4. In substitution, change everything, including the limits. Put u=g(x), replace du=g'(x)\,dx and rewrite the limits in u, as for \int2x\cos(x^2)\,dx=\sin(x^2)+C (Fig 13.2). If you keep the old limits with the new variable the value is wrong. The integrand must contain g'(x) as a factor.
5. By parts: \int u\,dv=uv-\int v\,du, with u chosen by LIATE. For \int xe^x\,dx take u=x and get (x-1)e^x+C. For \int\ln x\,dx take u=\ln x, dv=dx to get x\ln x-x+C. Choosing u=e^x makes the integral harder.
6. A definite integral is F(b)-F(a). Swapping the limits changes the sign. Split at a point where the function is undefined or changes rule. For an odd function \int_{-a}^{a}f\,dx=0, and \int_{-1}^{1}x^2dx=\frac23 for an even one.
7. Area is not the same as the signed integral. \int_0^{2\pi}\sin x\,dx=0 but the area between the curve and the axis is 4 (Fig 13.3). Split at the zeros and add the absolute values. The area between two curves is \int(\text{upper}-\text{lower})\,dx.
8. Memorise the standard forms and verify by differentiation. \int\sec^2x\,dx=\tan x, \int\frac{dx}{1+x^2}=\tan^{-1}x, \int\tan x\,dx=\ln|\sec x|, \int a^xdx=\frac{a^x}{\ln a} and \int e^xdx=e^x. The Riemann sum picture (Fig 13.1) is why the integral of x^2 on [0,2] is \frac83.

13.4.1 Worked checks

Q. Evaluate \int(3x^2+2x+1)\,dx and \int_0^2(3x^2+2x+1)\,dx.

Answer. The indefinite integral is x^3+x^2+x+C. The definite integral is [x^3+x^2+x]_0^2=8+4+2-0=14. The constant cancels in the definite case.

Q. Find \int xe^x\,dx and \int_0^1xe^x\,dx.

Answer. By parts, \int xe^xdx=xe^x-e^x+C=(x-1)e^x+C. Then \int_0^1=[(x-1)e^x]_0^1=0-(-1)=1. Differentiating (x-1)e^x gives xe^x, which confirms the result.

Q. Find the area enclosed by y=x and y=x^2.

Answer. They meet where x=x^2, at x=0 and x=1, and y=x is above. Area =\int_0^1(x-x^2)\,dx=\frac12-\frac13=\frac16. This is the shaded region in Fig 13.3.

Practice questions

Set A — Recall (20 questions)

1. \int x^2\,dx equalsax^3+Cb\frac{x^3}{3}+Cc2x+Cd\frac{x^2}{2}+C
2. \int\frac{1}{x}\,dx equalsax^0+Cb-\frac{1}{x^2}+Cc\frac{1}{x^2}+Cd\ln|x|+C
3. \int\cos x\,dx equalsa\sin x+Cb-\sin x+Cc-\cos x+Cd\cos x+C
4. \int e^x\,dx equalsaxe^x+Cbe^{x+1}+Cce^x+Cd\ln x+C
5. \int\sin x\,dx equalsa\cos x+Cb-\cos x+Cc\sin x+Cd-\sin x+C
6. \int_0^1x\,dx equalsa1b0c2d\frac{1}{2}
7. \int\sec^2x\,dx equalsa\sec x\tan x+Cb\cot x+Cc\tan x+Cd-\tan x+C
8. The integral of zero isaa constantb0cxd1
9. \int\frac{1}{1+x^2}\,dx equalsa\ln(1+x^2)+Cb\sin^{-1}x+Cc\frac{1}{x}+Cd\tan^{-1}x+C
10. \int_0^23\,dx equalsa3b6c9d2
11. \int(2x+3)\,dx equalsa2x^2+3x+Cbx^2+3+Ccx^2+3x+Cd2+C
12. \int_0^\pi\sin x\,dx equalsa2b0c1d-2
13. \int x^{-2}\,dx equalsa\frac{1}{x}+Cb-\frac{1}{x}+Cc-2x^{-3}+Cd\ln x+C
14. \int a^x\,dx equalsaa^x+Cba^x\ln a+Ccxa^{x-1}+Cd\frac{a^x}{\ln a}+C
15. \int_{-1}^{1}x^3\,dx equalsa0b\frac{1}{2}c2d\frac{1}{4}
16. \int\sqrt{x}\,dx equalsa\frac{1}{2}x^{-1/2}+Cbx^{3/2}+Cc\frac{2}{3}x^{3/2}+Cd2\sqrt{x}+C
17. \int_1^e\frac{1}{x}\,dx equalsaeb1c0de-1
18. \int\frac{f'(x)}{f(x)}\,dx equalsaf(x)^2+Cbf'(x)+Cc\frac{1}{f(x)}+Cd\ln|f(x)|+C
19. \int_0^{\pi/2}\cos x\,dx equalsa1b0c2d-1
20. The area under y=x from x=0 to x=4 isa4b16c8d2

Set B — Understanding (12 questions)

1. \int(3x^2+2x+1)\,dx equalsa6x+2+Cbx^3+x^2+Cc3x^3+x^2+x+Cdx^3+x^2+x+C
2. \int_1^22x\,dx equalsa3b4c2d1
3. \int\frac{2x}{x^2+1}\,dx equalsa2\ln x+Cbx^2+1+Cc\ln(x^2+1)+Cd\tan^{-1}x+C
4. \int_0^1e^x\,dx equalsaebe-1c1de+1
5. \int xe^x\,dx equalsa(x-1)e^x+Cb(x+1)e^x+Ccxe^x+Cd\frac{e^x}{2}+C
6. The area bounded by y=x^2, the x-axis and x=3 isa27b3c6d9
7. \int_0^2x^2\,dx equalsa4b\frac{8}{3}c\frac{2}{3}d8
8. \int\cos2x\,dx equalsa\sin2x+Cb2\sin2x+Cc\frac{\sin2x}{2}+Cd-\frac{\sin2x}{2}+C
9. \int_{-2}^{2}x^2\,dx equalsa\frac{16}{3}b0c\frac{8}{3}d8
10. \int\frac{dx}{2x+3} equalsa\ln|2x+3|+Cb2\ln|2x+3|+Cc-\frac{1}{(2x+3)^2}+Cd\frac{1}{2}\ln|2x+3|+C
11. \int_0^{\pi/4}\sec^2x\,dx equalsa0b1c\sqrt{2}d\frac{\pi}{4}
12. The area between y=x and y=x^2 from 0 to 1 isa\frac{1}{2}b\frac{1}{3}c\frac{1}{6}d\frac{1}{12}

Set C — Application (8 questions)

1. \int_0^1(x+1)^2\,dx equalsa3b\frac{7}{3}c\frac{4}{3}d\frac{1}{3}
2. \int_0^{\pi/2}\sin x\cos x\,dx equalsa1b\frac{1}{4}c0d\frac{1}{2}
3. \int_1^e\ln x\,dx equalsa1bece-1d0
4. \int_0^1\frac{dx}{\sqrt{1-x^2}} equalsa1b\pic\frac{\pi}{2}d\frac{\pi}{4}
5. The area under y=2x from x=0 to x=3 isa6b9c3d18
6. \int\frac{x^2+1}{x}\,dx equalsax^2+\ln|x|+Cb2x-\frac{1}{x^2}+Cc\frac{x^2}{2}+Cd\frac{x^2}{2}+\ln|x|+C
7. If \int_0^ax\,dx=8, then a isa2b8c4d16
8. The area enclosed by y=4-x^2 and the x-axis isa\frac{32}{3}b\frac{16}{3}c8d16