\(\int \frac{dx}{\sin^2 x \cos^2 x}\) equals
\(\int \frac{dx}{\sin^2 x \cos^2 x}\) equals
tan x + cot x +C
tan x - cot x +C
tan x cot x +C
tan x-cot 2x +C
The Correct Option is B
Solution and Explanation
Let\(I=\int \frac{dx}{\sin^2 x \cos^2 x}\)
= \(\int \frac{1}{\sin^2 x \cos^2 x}dx\)
= \(\int \frac{\sin^2 x+ \cos^2 x}{\sin^2 x \cos^2 x}dx\)
= \(\int \frac{\sin^2x}{\sin^2x\cos^2 x}dx+\int\frac{\cos^2 x}{\sin^2x\cos^2x}dx\)
= \(\int \sec^2 xdx+\int\cosec^2 xdx\)
= \(\tan x -\cot x+C\)
Hence, the correct answer is B.
Top Questions on integral
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Let \( f : (0, \infty) \to \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0 \), } \[ \int_0^a f(x) \, dx = f(a), \quad f(1) = 1, \quad f(16) = \frac{1}{8}, \quad \text{then } 16 - f^{-1}\left( \frac{1}{16} \right) \text{ is equal to:}\]
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Concepts Used:
Methods of Integration
Given below is the list of the different methods of integration that are useful in simplifying integration problems:
Integration by Parts:
If f(x) and g(x) are two functions and their product is to be integrated, then the formula to integrate f(x).g(x) using by parts method is:
∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C
Here f(x) is the first function and g(x) is the second function.
Method of Integration Using Partial Fractions:
The formula to integrate rational functions of the form f(x)/g(x) is:
∫[f(x)/g(x)]dx = ∫[p(x)/q(x)]dx + ∫[r(x)/s(x)]dx
where
f(x)/g(x) = p(x)/q(x) + r(x)/s(x) and
g(x) = q(x).s(x)
Integration by Substitution Method
Hence the formula for integration using the substitution method becomes:
∫g(f(x)) dx = ∫g(u)/h(u) du
Integration by Decomposition
Reverse Chain Rule
This method of integration is used when the integration is of the form ∫g'(f(x)) f'(x) dx. In this case, the integral is given by,
∫g'(f(x)) f'(x) dx = g(f(x)) + C