Ch 3 Maths Class 12 Solutions

• Important ex 3.1 class 12 maths for Intermediate part-II students.
• Integration: The technique or method to find such a function whose derivative is given involves the inverse process of differentiation called anit-derivation or integration.
• Important ex 3.2 class 12 maths for Intermediate part-II students.
• Integration as Anti-derivative (Inverse of derivative): The inverse process of differentiation i-e, the process of finding such a function whose derivative is given is called anti-differentiation or integration. 
• Important ex 3.3 class 12 maths for Intermediate part-II students.
• The symbol integral ...dx indicates that integrand is to be integrated with respect to "x".
• The antiderivative of a function is also called indefinite integral.
• The function which is to be integrated is called integrand of the integral.
• Important ex 3.4 class 12 maths for Intermediate part-II students.
• Integration by method of substitution: sometimes it is possible to convert an integral into standard form by a suitable change of a variable. This is called substitution method. i-e., Evaluate (f(x)dx by method of substitution. Let x = Φ(t) dx => Φ'(t)dt  so integral f(x)dx = (f(Φ(t) Φ'(t)dt
• Important ex 3.5 class 12 maths for Intermediate part-II students.
• "Some basic rules for Integration by parts"
• chose the function as 2nd function whose integration is known or possible.
• If integration of both given functions are known but one of the given function is polynomial function then chose polynomial function as Ist function.
• Important ex 3.6 class 12 maths for Intermediate part-II students.
• If integration of both given functions are known but no one is polynomial function then we may chose any function as 1st.
• If we are given only 1 function whose integration is unknown or cannot be easily find (i-e., sin'x, cos'x, Ja-x², J 1 then we take 1 as 2nd function. etc).
• Important ex 3.7 class 12 maths for Intermediate part-II students.
• You may remember the word "ILATE" I=Inverse functions, L = Logarithmic functions A= Algebraic functions, T= Trigonometric functions, E = Exponential functions. Integration involving "Partial Fractions"
• Important ex 3.8 class 12 maths for Intermediate part-II students.
• If P(x), Q(x) are two polynomial functions and Q(x)(0) in rational fraction P(x)/Q(x) Can be factorized in to linear and quadratic (irreducible) factors, then the rational function is written as a sum of simpler rational functions, each of which can be integrated by methods already known. Here we will discuss examples of the three cases of partial fraction and then apply integration.
• The Definit Integrals: If integrate f(x) dx = Phi(x) + c then the integral of fufrom denoted by Sf(x)dx and read as "definit integral of f(x). Here a is called Lower limit and bis called upper limit.
• ★The interval [a, b] is called range of integration
• We evaluat integrate f(x) dx from a to b as;
• Consider integrate f(x) dx = Phi(x) + c + integrate f(x) dx from a to b =| Phi(x)+ epsilon| a ^ b = [(b)+c]-[$(a)+c] = (b) + c - (a) - c C integrate f(x) dx from a to b = Phi(b) - Phi(a)
• Note: If the lower limit is a constant and upper limit is a variable, then the integral is a function of the upper limit {"f(t)dt=1$(t) = $(x)-中. 
• The Area Under the Curve integrate f(x) dx from a to b = Phi(b) - Phi(a) region" bounded under the curve of function f(x) the x-axis and between two ordinates x = a , x = b fig. as shown in Fundamental theorem of Calculus
• If f(x) is continuous forall x \in [a, b] and phi' * (x) = f(x) then integrate f(x) dx from a to b = Phi(b) - Phi(a) is called Fundamental integral calculus.
• Important ex 3.1 class 12 for Intermediate part-II students.
• Important ex 3.2 class 12 for Intermediate part-II students.
• Important ex 3.3 class 12 for Intermediate part-II students.
• Important ex 3.4 class 12 for Intermediate part-II students.
• Important ex 3.5 class 12 for Intermediate part-II students.
• Important ex 3.6 class 12 for Intermediate part-II students.
• Important ex 3.7 class 12 for Intermediate part-II students.
• Important ex 3.8 class 12 for Intermediate part-II students.