Quick answer: Integration by parts is ∫ u v d x = u ∫ v d x − ∫ ( u ′ ∫ v d x ) d x \int u\,v\,dx = u\int v\,dx - \int\left(u'\int v\,dx\right)dx ∫ u v d x = u ∫ v d x − ∫ ( u ′ ∫ v d x ) d x ILATE order (Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential). A useful shortcut is ∫ e x [ f ( x ) + f ′ ( x ) ] d x = e x f ( x ) + C \int e^x[f(x) + f'(x)]\,dx = e^xf(x) + C ∫ e x [ f ( x ) + f ′ ( x )] d x = e x f ( x ) + C Partial fractions split a rational function into simpler fractions, each with a standard integral such as ∫ d x x 2 − a 2 = 1 2 a ln ∣ x − a x + a ∣ + C \int\frac{dx}{x^2 - a^2} = \frac{1}{2a}\ln\left\lvert\frac{x - a}{x + a}\right\rvert + C ∫ x 2 − a 2 d x = 2 a 1 ln x + a x − a + C
This continues from indefinite integrals and substitution .
Integration by parts
∫ u v d x = u ∫ v d x − ∫ ( d u d x ∫ v d x ) d x \int u\,v\,dx = u\int v\,dx - \int\left(\frac{du}{dx}\int v\,dx\right)dx ∫ u v d x = u ∫ v d x − ∫ ( d x d u ∫ v d x ) d x
ILATE rule: choose u as the function that comes first in the list.
Letter Type Example I Inverse trigonometric tan − 1 x \tan^{-1}x tan − 1 x L Logarithmic ln x \ln x ln x A Algebraic x 2 x^2 x 2 T Trigonometric sin x \sin x sin x E Exponential e x e^x e x
Useful results by parts
Integral Result (+ C) ∫ ln x d x \int\ln x\,dx ∫ ln x d x x ln x − x x\ln x - x x ln x − x ∫ x e x d x \int xe^x\,dx ∫ x e x d x ( x − 1 ) e x (x - 1)e^x ( x − 1 ) e x ∫ e x [ f ( x ) + f ′ ( x ) ] d x \int e^x[f(x) + f'(x)]\,dx ∫ e x [ f ( x ) + f ′ ( x )] d x e x f ( x ) e^xf(x) e x f ( x ) ∫ e a x sin b x d x \int e^{ax}\sin bx\,dx ∫ e a x sin b x d x e a x a 2 + b 2 ( a sin b x − b cos b x ) \frac{e^{ax}}{a^2 + b^2}(a\sin bx - b\cos bx) a 2 + b 2 e a x ( a sin b x − b cos b x ) ∫ e a x cos b x d x \int e^{ax}\cos bx\,dx ∫ e a x cos b x d x e a x a 2 + b 2 ( a cos b x + b sin b x ) \frac{e^{ax}}{a^2 + b^2}(a\cos bx + b\sin bx) a 2 + b 2 e a x ( a cos b x + b sin b x )
Special integrals
Integral Result (+ C) ∫ d x x 2 + a 2 \int\frac{dx}{x^2 + a^2} ∫ x 2 + a 2 d x 1 a tan − 1 x a \frac{1}{a}\tan^{-1}\frac{x}{a} a 1 tan − 1 a x ∫ d x x 2 − a 2 \int\frac{dx}{x^2 - a^2} ∫ x 2 − a 2 d x 1 2 a ln ∣ x − a x + a ∣ \frac{1}{2a}\ln\left\lvert\frac{x - a}{x + a}\right\rvert 2 a 1 ln x + a x − a ∫ d x a 2 − x 2 \int\frac{dx}{a^2 - x^2} ∫ a 2 − x 2 d x 1 2 a ln ∣ a + x a − x ∣ \frac{1}{2a}\ln\left\lvert\frac{a + x}{a - x}\right\rvert 2 a 1 ln a − x a + x ∫ d x a 2 − x 2 \int\frac{dx}{\sqrt{a^2 - x^2}} ∫ a 2 − x 2 d x sin − 1 x a \sin^{-1}\frac{x}{a} sin − 1 a x ∫ d x x 2 + a 2 \int\frac{dx}{\sqrt{x^2 + a^2}} ∫ x 2 + a 2 d x ln ∣ x + x 2 + a 2 ∣ \ln\left\lvert x + \sqrt{x^2 + a^2}\right\rvert ln x + x 2 + a 2 ∫ d x x 2 − a 2 \int\frac{dx}{\sqrt{x^2 - a^2}} ∫ x 2 − a 2 d x ln ∣ x + x 2 − a 2 ∣ \ln\left\lvert x + \sqrt{x^2 - a^2}\right\rvert ln x + x 2 − a 2 ∫ a 2 − x 2 d x \int\sqrt{a^2 - x^2}\,dx ∫ a 2 − x 2 d x x 2 a 2 − x 2 + a 2 2 sin − 1 x a \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} 2 x a 2 − x 2 + 2 a 2 sin − 1 a x
For ∫ d x a x 2 + b x + c \int\frac{dx}{ax^2 + bx + c} ∫ a x 2 + b x + c d x
Partial fractions
For a proper rational function (degree of numerator less than the denominator):
Denominator factor Partial fraction form ( x − a ) (x - a) ( x − a ) A x − a \frac{A}{x - a} x − a A ( x − a ) 2 (x - a)^2 ( x − a ) 2 A x − a + B ( x − a ) 2 \frac{A}{x - a} + \frac{B}{(x - a)^2} x − a A + ( x − a ) 2 B ( x 2 + b x + c ) (x^2 + bx + c) ( x 2 + b x + c ) A x + B x 2 + b x + c \frac{Ax + B}{x^2 + bx + c} x 2 + b x + c A x + B
If the fraction is improper, divide first.
Cover-up rule: for a non-repeated linear factor ( x − a ) (x - a) ( x − a ) x = a x = a x = a
Worked examples
Example 1: ILATE
Evaluate ∫ x cos x d x \int x\cos x\,dx ∫ x cos x d x
u = x, v = cos x: x sin x − ∫ sin x d x = x sin x + cos x + C x\sin x - \int\sin x\,dx = x\sin x + \cos x + C x sin x − ∫ sin x d x = x sin x + cos x + C
Example 2: log by parts
Evaluate ∫ x ln x d x \int x\ln x\,dx ∫ x ln x d x
u = ln x, v = x:
x 2 2 ln x − ∫ x 2 2 ⋅ 1 x d x = x 2 2 ln x − x 2 4 + C \frac{x^2}{2}\ln x - \int\frac{x^2}{2} \cdot \frac{1}{x}\,dx = \frac{x^2}{2}\ln x - \frac{x^2}{4} + C 2 x 2 ln x − ∫ 2 x 2 ⋅ x 1 d x = 2 x 2 ln x − 4 x 2 + C
Example 3: eˣ shortcut
Evaluate ∫ e x ( sin x + cos x ) d x \int e^x(\sin x + \cos x)\,dx ∫ e x ( sin x + cos x ) d x
Here f = sin x f = \sin x f = sin x f ′ = cos x f' = \cos x f ′ = cos x e x sin x + C e^x\sin x + C e x sin x + C
Example 4: partial fractions
Evaluate ∫ d x ( x − 1 ) ( x + 2 ) \int\frac{dx}{(x - 1)(x + 2)} ∫ ( x − 1 ) ( x + 2 ) d x
Step 1: By cover-up, A = 1 3 A = \frac{1}{3} A = 3 1 x = 1 x = 1 x = 1 B = − 1 3 B = -\frac{1}{3} B = − 3 1 x = − 2 x = -2 x = − 2
Step 2:
1 3 ln ∣ x − 1 ∣ − 1 3 ln ∣ x + 2 ∣ + C = 1 3 ln ∣ x − 1 x + 2 ∣ + C \frac{1}{3}\ln\lvert x - 1\rvert - \frac{1}{3}\ln\lvert x + 2\rvert + C = \frac{1}{3}\ln\left\lvert\frac{x - 1}{x + 2}\right\rvert + C 3 1 ln ∣ x − 1 ∣ − 3 1 ln ∣ x + 2 ∣ + C = 3 1 ln x + 2 x − 1 + C
Example 5: completing the square
Evaluate ∫ d x x 2 + 4 x + 13 \int\frac{dx}{x^2 + 4x + 13} ∫ x 2 + 4 x + 13 d x
x 2 + 4 x + 13 = ( x + 2 ) 2 + 9 x^2 + 4x + 13 = (x + 2)^2 + 9 x 2 + 4 x + 13 = ( x + 2 ) 2 + 9 1 3 tan − 1 x + 2 3 + C \frac{1}{3}\tan^{-1}\frac{x + 2}{3} + C 3 1 tan − 1 3 x + 2 + C
Summary table
Technique Use when By parts Product of two different types of function ILATE Choosing u e x [ f + f ′ ] e^x[f + f'] e x [ f + f ′ ] eˣ times a function plus its derivative Partial fractions Rational functions with factorable denominators Completing the square Quadratic denominators
Exam tips
Look for the e x [ f + f ′ ] e^x[f + f'] e x [ f + f ′ ]
Use the cover-up rule for quick partial fraction constants.
For ∫ ln x d x \int\ln x\,dx ∫ ln x d x ∫ tan − 1 x d x \int\tan^{-1}x\,dx ∫ tan − 1 x d x
Next, read definite integrals and their properties , or see the NIMCET Calculus chapter .
Common mistakes
Choosing u against the ILATE order, making the integral harder.
Forgetting the minus sign in the parts formula.
Applying partial fractions to an improper fraction without dividing first.
Mixing up 1 x 2 − a 2 \frac{1}{x^2 - a^2} x 2 − a 2 1 1 a 2 − x 2 \frac{1}{a^2 - x^2} a 2 − x 2 1
Frequently Asked Questions
∫uv dx = u∫v dx − ∫(u′∫v dx) dx.
What is the ILATE rule?
An order for choosing the first function u: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential.
What is the integral of eˣ[f(x) + f′(x)]?
eˣf(x) + C.
What is the integral of 1/(x² + a²)?
(1/a) tan⁻¹(x/a) + C.
What is the integral of ln x?
x ln x − x + C.
