Find the integrals of the function: \(tan^4x\)
Solution and Explanation
tan4x
=tan2 x. tan2 x
=(sec2 x-1)tan2 x
=sec2 x tan2 x-tan2 x
=sec2 x tan2 x-(sec2 x-1)
= sec2 x tan2 x-sec2 x+1
∴ ∫tan4 x dx = ∫sec2 xtan2 x dx- ∫sec2 x dx+ ∫1.dx
= ∫sec2 x tan2 x dx-tan x +x+C ...(1)
Consider ∫sec2 x tan2 x dx
Let tan x = t ⇒ sec2 x dx = dt
⇒ ∫sec2 x tan2 xdx = ∫t2dt \(= \frac{t^3}{3} = \frac{tan^3x}{3}\)
From equation (1), we obtain
\(∫tan^4 x dx = \frac{1}{3} tan^3 x-tan x +x+C\)
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Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
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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