Integrals represent the accumulation of quantities and the area under curves. They are classified into definite and indefinite integrals. The definite integral computes a specific area, while the indefinite integral finds the antiderivative. Integration is the reverse process of differentiation and is fundamental in calculus for solving various mathematical and physical problems.
Class 12 Maths Chapter 7 Integrals is an important topic for the CBSE Exam 2024-25. It deals with finding the area under curves and solving accumulation problems. There are two types of integrals definite and indefinite. Definite integrals compute a specific value and indefinite integrals find antiderivatives which are useful in physics and engineering.
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We are given that
d/dx [f(x)] = ax + b
Step 1: Integrate both sides
To find f(x), integrate the given derivative:
f(x) = ∫ (ax + b) dx
Using standard integration rules:
∫ ax dx = (a x²)/2 and ∫ b dx = bx
Thus,
f(x) = (a x²)/2 + bx + C
where C is the constant of integration.
Step 2: Use the given condition f(0) = 0
Substituting x = 0 in the equation:
0 = (a(0)²)/2 + b(0) + C
0 = C
Thus, C = 0, so the final function is:
f(x) = (a x²)/2 + bx
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