Integration by Parts – Problems Integrals of the form ∫eax sin bx dx and ∫eax cos bx dx In order to evaluate this type of integrals, we use integration by parts as illustrated below. I = ∫eax sin bx dx , , , , , ∴ , , , Thus , Similarly, . Example: ∫sin (log Read more about Integration by Parts – Problems[…]

Integration by Parts If u and v are two functions of x, then i.e., the integral of the product of two functions = (first function) x (integral of second function) – integral of {diff of first function) x (integral of second function)} we can choose the first function which comes first in the word ILATE, where Read more about Integration by Parts[…]

Fundamental Theorem of Calculus The fundamental theorem of calculus is a theorem that likes the concept of the derivative of a function with the concept of the integral. (i) First fundamental theorem of calculus: Let f be a continuous real valued function defined on a closed interval [a, b]. let F be the function defined, Read more about Fundamental Theorem of Calculus[…]

Integral Function and its Properties Let f(x) be a continuous function defined on [a, b], then a function φ(x) defined by φ (x) = a∫x f (t) dt for all x ϵ [a, b] is called the integral function of the function f(x) Property I: The integral function of an integrable function is always continuous. Read more about Integral Function and its Properties[…]

Linear Differential Equation A differential equation is said to be linear. If the dependent variable and its differential coefficients occur in the first degree only and are not multiplied together. Method 1: The most general from of a linear equation of the first order is , Where, P and Q are any function of x. Read more about Linear Differential Equation[…]

Differentiation – Implicit Function i) In given implicit function, we solve this for dy/dx directly ii) Moderate level sometime in question, we asked dy/ dx at some particular point x. So, after getting dy/dx, we put the given particular value of x in dy/ dx and get the desired result. If the relation between the Read more about Differentiation – Implicit Function[…]

Differentiation – Logarithmic The function which can be evaluate by using logarithmic differentiation are of type y = [f (x)]g(x). y = f₁ (x). f₂ (x). f₃ (x) … . For this type of function take the logarithm and then differentiation. Example 1: If xy. yx = 1, then the value of =? Solution: Given that, Read more about Differentiation – Logarithmic[…]

Differentiation of Determinants If Then, i.e., to differentiate a determinant, differentiate one row (or column) at a time, keeping other unchanged. Example: If , then f’(x) is equal to Solution: ∵ ∴ = 1(12x² – 6x²) + 6x (2x – 2x) – 2(6x² – 3x²) + 6 (2x² – x²) = 6x² + 0 – Read more about Differentiation of Determinants[…]

Differentiation – Rules 2 Derivative of Composite Function: If f(u) is differentiable at the point u = g(x) and g(x) is differentiable at x, then the composite function (fog)(x) = f{g(x)} is differentiable at x and (fog)’(x) = f’{g(x)}. g’(x). Relation between dy/dx and dx/dy: If the inverse functions f and g are defined by Read more about Differentiation – Rules 2[…]

Differentiation – Rules Product Rule of Differentiation: The derivative of the product of two functions . = (First Function) x (Derivative of Second Function) + (Second Function) x (Derivative of First Function) Quotient Rule of Differentiation: The derivative of the quotient of two functions . . Derivative of a Function: (Chain Rule) If y is Read more about Differentiation – Rules[…]