JEE MainMathematicsEasyDefinite Integration

The Value
$I = \int_{- 1}^{1}\mspace{2mu}\left( \cos^{- 1}x + \frac{x^{7} - 3x^{5} + 7x^{3} - x}{\cos^{2}x} \right)dx$
is

Show answer and solution

Let

f(x) = \frac{x^{7} - 3x^{5} + 7x^{3} - x}{\cos^{2}x}

then

f( - x) = - f(x)

so,

\int_{- 1}^{1}\mspace{2mu} f(x)dx = 0

(Property
12)
Let

g(x) = \cos^{- 1}x\

then

I_{1} = \int_{- 1}^{1}\mspace{2mu}\cos^{- 1}xdx

= \int_{- 1}^{1}\mspace{2mu}\cos^{- 1}( - x)dx

(Property 8)
Thus I = I_1 =

\pi

Source: Grass-G / M-13 Mathematics Paper 4 Nov.docx

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