Integration by parts
When two people pass a heavy box, one holds and the other moves it. In a product, one factor is differentiated and the other is integrated, and the problem becomes lighter.
For a product of two functions, choose a first function u and a second function v. Differentiate u and integrate v. The order for the first function is ILATE: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential.
Formula: ∫ u v dx = u ∫ v dx − ∫ [u′ (∫ v dx)] dx
Rule 1: Choose by ILATE — Pick u as the function that comes first in the list.
- Easy: ∫ x eˣ dx = x eˣ − eˣ + C.
- Medium: ∫ x cos x dx = x sin x + cos x + C.
- Exam-type: ∫ ln x dx = ∫ (ln x) · 1 dx = x ln x − x + C.
Rule 2: Repeat if needed — The power of x drops by one in every round.
- Easy: ∫ x² eˣ dx = eˣ (x² − 2x + 2) + C.
- Medium: ∫ x ln x dx = (x²/2) ln x − x²/4 + C.
- Exam-type: ∫ tan⁻¹x dx = x tan⁻¹x − ½ ln(1 + x²) + C.
Check the answer by differentiating. The result must be the original integrand.
Choosing eˣ as the first function in ∫ x eˣ dx, which makes the algebraic power grow instead of drop.
ILATE, and the first letter wins.
Practice questions
- 1Find ∫ x eˣ dx.
- 2Find ∫ x sin x dx.
- 3Find ∫ ln x dx.
- 4Find ∫ x ln x dx.
- 5Find ∫ x² eˣ dx.
- 6Find ∫ x e^(2x) dx.
- 7Find ∫ sin⁻¹x dx.
- 8Find ∫ x² cos x dx.
Show answers (try them yourself first)
- 1x eˣ − eˣ + C
- 2sin x − x cos x + C
- 3x ln x − x + C
- 4(x²/2) ln x − x²/4 + C
- 5eˣ (x² − 2x + 2) + C
- 6e^(2x) (x/2 − 1/4) + C
- 7x sin⁻¹x + √(1 − x²) + C
- 8x² sin x + 2x cos x − 2 sin x + C
Integrals of the type eˣ [f(x) + f′(x)]
A wrapped gift has the surprise inside only when both the function and its slope are present. If a function and its derivative sit together beside eˣ, the integral is found at once.
When the integrand is eˣ times the sum of a function and its derivative, the integral is simply eˣ times that function.
Formula: ∫ eˣ [f(x) + f′(x)] dx = eˣ f(x) + C
Rule 1: Spot f and f′ — Check that the second part is the derivative of the first.
- Easy: ∫ eˣ (sin x + cos x) dx = eˣ sin x + C.
- Medium: ∫ eˣ (1/x − 1/x²) dx = eˣ/x + C.
- Exam-type: ∫ eˣ (1 + sin x)/(1 + cos x) dx = eˣ tan(x/2) + C.
Rule 2: Hidden form — Expand or rearrange so that f and f′ appear.
- Easy: ∫ eˣ (x + 1) dx = x eˣ + C.
- Medium: ∫ eˣ (tan x + sec²x) dx = eˣ tan x + C.
- Exam-type: ∫ eˣ · x/(x + 1)² dx = ∫ eˣ [1/(x + 1) − 1/(x + 1)²] dx = eˣ/(x + 1) + C.
The answer is eˣ times the first part only, not the sum.
Giving eˣ (f + f′) as the answer, or missing that the pair must match exactly.
Function and slope together, e to the x times the function.
Practice questions
- 1Find ∫ eˣ (sin x + cos x) dx.
- 2Find ∫ eˣ (x + 1) dx.
- 3Find ∫ eˣ (1/x − 1/x²) dx.
- 4Find ∫ eˣ (tan x + sec²x) dx.
- 5Find ∫ eˣ (x² + 2x) dx.
- 6Find ∫ eˣ (cos x − sin x) dx.
- 7Find ∫ eˣ (1 + sin x)/(1 + cos x) dx.
- 8Find ∫ eˣ · x/(x + 1)² dx.
Show answers (try them yourself first)
- 1eˣ sin x + C
- 2x eˣ + C
- 3eˣ/x + C
- 4eˣ tan x + C
- 5x² eˣ + C
- 6eˣ cos x + C
- 7eˣ tan(x/2) + C
- 8eˣ/(x + 1) + C
Integrals of some more types
A swing that goes back and forth keeps returning to the same shape. Integrals of eᵃˣ with sin bx or cos bx return to themselves after two rounds of parts, and then a small equation gives the answer.
For ∫ eᵃˣ sin bx dx, apply parts twice. The original integral appears again on the right side. Move it to the left and solve. The result is a standard formula.
Formula: ∫ eᵃˣ sin bx dx = eᵃˣ (a sin bx − b cos bx)/(a² + b²) + C, ∫ eᵃˣ cos bx dx = eᵃˣ (a cos bx + b sin bx)/(a² + b²) + C
Rule 1: Use the formulas — Put in the values of a and b.
- Easy: ∫ eˣ sin x dx = eˣ (sin x − cos x)/2 + C.
- Medium: ∫ e^(2x) cos x dx = e^(2x) (2 cos x + sin x)/5 + C.
- Exam-type: ∫ e^(3x) sin 2x dx = e^(3x) (3 sin 2x − 2 cos 2x)/13 + C.
Rule 2: Negative or definite — a may be negative, and limits are put in at the end.
- Easy: ∫ e^(−x) sin x dx = −e^(−x) (sin x + cos x)/2 + C.
- Medium: ∫ from 0 to π of eˣ sin x dx = (e^π + 1)/2.
- Exam-type: ∫ e^(−2x) cos x dx = e^(−2x) (−2 cos x + sin x)/5 + C.
The denominator is a² + b². The signs of the two terms in the bracket differ for sine and cosine.
Mixing up the bracket of the sine formula with the cosine formula.
Sine: a sin minus b cos. Cosine: a cos plus b sin.
Practice questions
- 1Find ∫ eˣ cos x dx.
- 2Find ∫ e^(2x) sin x dx.
- 3Find ∫ e^(−x) sin x dx.
- 4Find ∫ eˣ sin 2x dx.
- 5Find ∫ e^(2x) cos 3x dx.
- 6Find ∫ e^(3x) cos x dx.
- 7Find ∫ e^(−2x) cos x dx.
- 8Find the value of ∫ from 0 to π of eˣ sin x dx.
Show answers (try them yourself first)
- 1eˣ (cos x + sin x)/2 + C
- 2e^(2x) (2 sin x − cos x)/5 + C
- 3−e^(−x) (sin x + cos x)/2 + C
- 4eˣ (sin 2x − 2 cos 2x)/5 + C
- 5e^(2x) (2 cos 3x + 3 sin 3x)/13 + C
- 6e^(3x) (3 cos x + sin x)/10 + C
- 7e^(−2x) (−2 cos x + sin x)/5 + C
- 8(e^π + 1)/2