An anti-derivative is a function whose derivative gives the original function. It is also called an indefinite integral. Finding the anti-derivative is the reverse process of differentiation. It includes a constant of integration since differentiation removes constants. Anti-derivatives are essential in calculus for solving area, motion, and accumulation problems.
Class 12 Maths Chapter 7 Integrals is an essential topic for the CBSE Exam 2024-25. It focuses on finding areas under curves and solving accumulation problems. There are two types of integrals definite and indefinite. Definite integrals compute exact values and indefinite integrals determine antiderivatives which are widely used in physics and engineering.
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To determine the antiderivative (indefinite integral) of
I = ∫ (tan x – 1) / (tan x + 1) dx
Step 1: Substituting x in terms of a trigonometric identity
We apply the identity :
tan(A – B) = (tan A – tan B) / (1 + tan A tan B)
Here, we choose A = π/4 and B = x, so
tan(π/4 – x) = (tan(π/4) – tan x) / (1 + tan(π/4) tan x)
Since tan(π/4) = 1, this reduces to:
tan(π/4 – x) = (1 – tan x) / (1 + tan x)
Taking the negative,
– tan(π/4 – x) = (tan x – 1) / (tan x + 1)
So, our integral is:
I = ∫ – tan(π/4 – x) dx
Step 2: Finding the Integral
We know:
∫ tan u du = log | sec u | + C
Substituting u = π/4 – x, we get:
I = – ∫ tan(π/4 – x) dx
= – log | sec(π/4 – x) | + C
Conclusion
Therefore, the right answer is: – log | sec(π/4 – x) | + C
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