f ( x ) = 6 s i n − 1 ( 1 − x 2 ) = { 6 c o s − 1 ( x ) if − 1 ≤ x ≤ 0 6 π − 6 c o s − 1 ( x ) if 0 ≤ x ≤ 1 Domain of f ( x ) is : x ∈ [ − 1 , 1 ] Range of f ( x ) is : f ( x ) ∈ [ 0 , 3 π ] f ′ ( x ) = { − 3 1 − x 2 < 0 if − 1 ≤ x < 0 + 3 1 − x 2 > 0 if 0 < x ≤ 1 f(x)=6sin^{-1}(\sqrt{1-x^2})\\
~~~~~~~~~= \begin{cases}
6cos^{-1}(x) &\text{if }-1\leq x\leq0 \\
6\pi-6cos^{-1}(x) &\text{if }\;0\leq x\leq1
\end{cases}\\\;\\
\text{Domain of }f(x) \;\text{is}: x\in [-1,1]\\\;\\
\text{Range of }f(x)\;\text{is} : f(x)\in \left[0,3\pi\right]\\\;\\
f'(x)= \begin{cases}
-\dfrac{3}{\sqrt{1-x^2}}<0 &\text{if }-1\leq x<0 \\
+\dfrac{3}{\sqrt{1-x^2}}>0 &\text{if }\;0< x\leq1
\end{cases} f ( x ) = 6 s i n − 1 ( 1 − x 2 ) = { 6 co s − 1 ( x ) 6 π − 6 co s − 1 ( x ) if − 1 ≤ x ≤ 0 if 0 ≤ x ≤ 1 Domain of f ( x ) is : x ∈ [ − 1 , 1 ] Range of f ( x ) is : f ( x ) ∈ [ 0 , 3 π ] f ′ ( x ) = ⎩ ⎨ ⎧ − 1 − x 2 3 < 0 + 1 − x 2 3 > 0 if − 1 ≤ x < 0 if 0 < x ≤ 1
f ( x ) is not differentiable at x = 0 f(x) \text{ is not differentiable at } x=0 f ( x ) is not differentiable at x = 0
Now, ∫ c o s − 1 ( x ) d x = x c o s − 1 ( x ) − 1 − x 2 + C Area bounded by the curve and the x-axis is :- ∫ 1 − 1 f ( x ) d x = 2 × ∫ 0 1 f ( x ) d x = 2 × 6 = 12 \text{Now, }\\
{\displaystyle\int}cos^{-1}\left(x\right)\,\mathrm{d}x
=xcos^{-1}\left(x\right)-\sqrt{1-x^2}+C\\
\text{Area bounded by the curve and the x-axis is :-}\\
{{\displaystyle\int}_{1}^{-1}{f(x)}\,\mathrm{d}x =2\times {\displaystyle\int}^1_{0}{f(x)}\,\mathrm{d}x=2\times6=12} Now, ∫ co s − 1 ( x ) d x = x co s − 1 ( x ) − 1 − x 2 + C Area bounded by the curve and the x-axis is :- ∫ 1 − 1 f ( x ) d x = 2 × ∫ 0 1 f ( x ) d x = 2 × 6 = 12
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