Formula sheet: Integrals

Integrals14 min readLesson 1 of 8

1.1 What an integral is Integration reverses differentiation. If the derivative of F is f, then F is an antiderivative of f and the indefinite integral of f is F plus a constant. The constant matters because every antiderivative of a function differs from every other by a constant, and an extra condition such as a value of the function at one point fixes that constant. A definite integral is a number: it is the difference of an antiderivative evaluated at the upper and lower limits. The two ideas are linked by the fundamental theorem of calculus, which turns the problem of finding areas into the problem of finding antiderivatives. 1.2 Standard integrals The table below is the base of the chapter. Every method later in the chapter, substitution, partial fractions or by parts, only converts an unfamiliar integral into one of these forms. Learn the power rule with its exception at n equal to minus one, the exponential and logarithmic forms, and the six circular forms with the two inverse tangent and inverse sine forms.

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C (n \neq = - 1), \int \frac{1}{x} d x = lo g ∣ x ∣ + C

Power rule and the logarithmic case

\int e^{x} d x = e^{x} + C, \int a^{x} d x = \frac{a ^{x}}{lo g a} + C

Exponential integrals

\int sin x d x = - cos x + C, \int cos x d x = sin x + C

Sine and cosine

\int sec^{2} x d x = tan x + C, \int csc^{2} x d x = - cot x + C

Squares of secant and cosecant

\int sec x tan x d x = sec x + C, \int csc x cot x d x = - csc x + C

Products with tangent and cotangent

\int \frac{d x}{1 + x ^{2}} = tan^{- 1} x + C, \int \frac{d x}{1 - x ^{2}} = sin^{- 1} x + C

The two inverse circular forms

\int \frac{d x}{x ^{2} + a ^{2}} = \frac{1}{a} tan^{- 1} \frac{x}{a} + C, \int \frac{d x}{a ^{2} - x ^{2}} = sin^{- 1} \frac{x}{a} + C

Scaled inverse forms with a positive constant a

1.3 Methods and definite integrals The substitution rule reverses the chain rule, the by parts rule reverses the product rule, and partial fractions split a rational function into pieces that the table can handle. For a definite integral the properties below reduce the work: reversing the limits changes the sign, splitting at an intermediate point, and replacing x by a minus x are the three used most often in board questions. The limit of a sum form gives the same definite integral as a limit and is the definition used when the antiderivative cannot be found directly.

\int u v^{'} d x = u v - \int u^{'} v d x

Integration by parts; choose u by the LIATE order

\int_{a}^{b} f (x) d x = F (b) - F (a)

Fundamental theorem of calculus, with F an antiderivative of f

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, \int_{a}^{a} f (x) d x = 0

Reversing the limits and equal limits

\int_{a}^{b} f (x) d x = \int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x

Splitting the interval at c

\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x

The substitution that kills many symmetric integrals

\int_{a}^{b} f (x) d x = h \to 0 lim h [f (a) + f (a + h) + \dots + f (a + (n - 1) h)], nh = b - a

Definite integral as the limit of a sum

Exam

Write the substitution or the parts choice on the first line, then show the transformed integral. In a definite integral, change the limits when the variable is changed, or convert back before substituting the original limits; mixing the two is the most common source of lost marks.

Check whether the integrand is a standard form before trying a method.
For substitution, change both the integrand and dx, and change the limits for a definite integral.
For by parts, choose u in the LIATE order: inverse, logarithmic, algebraic, trigonometric, exponential.
For a rational function, factor the denominator and set up partial fractions before integrating.
In a definite integral, check the odd and even symmetry before expanding anything.

1.4 The order to try the methods Scan the integrand from left to right: a standard form is integrated directly; a product with a function and its derivative invites substitution; a rational function invites partial fractions; a product of unrelated types invites by parts. For a definite integral, test the symmetry properties first, because they can remove the work entirely. This order is the same in the exercise sets and in the board questions.

in hinglish

Integrals differentiation ka ulta hai; antiderivative ke saath +C likhna zaroori hai aur definite integral ka answer ek number hota hai. Standard integrals yaad rakho: power rule, 1/x ka log, e^x, sin, cos, sec^2, tan^-1 aur sin^-1 forms. Substitution chain rule ka reverse hai; definite integral mein substitution karo to limits bhi badlo. By parts product rule ka reverse hai, aur u ka choice LIATE order se karo. Rational function ke liye partial fractions use karo. Definite integral ki properties se kaafi questions aasan ho jaate hain, jaise f(x) ko f(a-x) se replace karna. Limits ka sign aur symmetry check karna mat bhoolo. Aur yaad rakho, definite integral mein limits reverse karne par sign badalta hai aur substitution ke saath limits bhi badalni padti hain. Answer milne ke baad differentiate karke check karna sabse aasan verification hai.