The value of the integral `inte^(3sin^(-1)x)((1)/(sqrt(1-x^(2)))+e^(3cos^(-1)x))dx` is equal to
(where, c is an arbitrary constant)

Options

A. `(e^(3sqrt(sin^(-1)x)))/(3)+xe^((3pi)/(2))+c`
B. `e^(sqrt(sin^(-1)x))+e^(pi//2)+c`
C. `(e^(3sin^(-1)x))/(3)+xe^((3pi)/(2))+c`
D. `e^((pi)/(2))+e^(x)((pi)/(2))+c`

Answer

To solve the integral

\[ \int e^{3 \sin^{-1} x} \left( \frac{1}{\sqrt{1 - x^2}} + e^{3 \cos^{-1} x} \right) dx, \]

we can break it down step by step.

### Step 1: Rewrite the Integral

We start by rewriting the integral:

\[ \int e^{3 \sin^{-1} x} \left( \frac{1}{\sqrt{1 - x^2}} + e^{3 \cos^{-1} x} \right) dx = \int e^{3 \sin^{-1} x} \frac{1}{\sqrt{1 - x^2}} \, dx + \int e^{3 \sin^{-1} x} e^{3 \cos^{-1} x} \, dx. \]

### Step 2: Simplify the Second Integral

Using the identity \( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \), we can simplify the second integral:

\[ e^{3 \cos^{-1} x} = e^{3 \left( \frac{\pi}{2} - \sin^{-1} x \right)} = e^{\frac{3\pi}{2}} e^{-3 \sin^{-1} x}. \]

Thus, the second integral becomes:

\[ \int e^{3 \sin^{-1} x} e^{3 \cos^{-1} x} \, dx = e^{\frac{3\pi}{2}} \int dx. \]

### Step 3: Combine the Integrals

Now we can combine the two integrals:

\[ \int e^{3 \sin^{-1} x} \frac{1}{\sqrt{1 - x^2}} \, dx + e^{\frac{3\pi}{2}} \int dx. \]

### Step 4: Change of Variables

Let \( t = \sin^{-1} x \). Then, we have:

\[ dx = \cos(t) \, dt = \sqrt{1 - \sin^2(t)} \, dt = \sqrt{1 - x^2} \, dt. \]

Thus, the first integral becomes:

\[ \int e^{3t} \, dt. \]

### Step 5: Evaluate the Integrals

1. For the first integral:

\[ \int e^{3t} \, dt = \frac{1}{3} e^{3t} + C_1 = \frac{1}{3} e^{3 \sin^{-1} x} + C_1. \]

2. For the second integral:

\[ e^{\frac{3\pi}{2}} \int dx = e^{\frac{3\pi}{2}} x + C_2. \]

### Step 6: Combine Results

Combining both results, we have:

\[ \int e^{3 \sin^{-1} x} \left( \frac{1}{\sqrt{1 - x^2}} + e^{3 \cos^{-1} x} \right) dx = \frac{1}{3} e^{3 \sin^{-1} x} + e^{\frac{3\pi}{2}} x + C, \]

where \( C = C_1 + C_2 \) is the arbitrary constant.

### Final Answer

Thus, the value of the integral is:

\[ \frac{1}{3} e^{3 \sin^{-1} x} + e^{\frac{3\pi}{2}} x + C. \]