If ∫ex2−x2(1−x)1−x2dx=μex1+x1−xλ+C then the value of 2(λ+μ) is

Solution:

Let $I = \int \frac{e^{x} (2 - x^{2})}{(1 - x) \sqrt{1 - x^{2}}} d x$ then

$\begin{array}{ll} \Rightarrow I = \int e^{x} (\frac{1 + (1 - x^{2})}{(1 - x) \sqrt{1 - x^{2}}}) d x \\ = \int e^{x} \{\frac{1}{(1 - x) \sqrt{1 - x^{2}}} + \sqrt{\frac{1 + x}{1 - x}}\} d x \\ \Rightarrow I = e^{x} \sqrt{\frac{1 + x}{1 - x}} + C \\ ∴ λ = \frac{1}{2} and μ = 1 \end{array}$

Hence, $2 (λ + μ) = 3$

If $\int \frac{e^{x} (2 - x^{2})}{(1 - x) \sqrt{1 - x^{2}}} d x = μ e^{x} {(\frac{1 + x}{1 - x})}^{λ} + C$ then the value of $2 (λ + μ)$ is

Solution:

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