The value of `int_(-1)^(1)(sin^(-1)x+(x^(5)+x^(3)-1)/(cos^(2)x))dx` is equal to

Options

A. `tan1`
B. 0
C. `2tan1`
D. `-2tan1`

Answer

To solve the integral

\[ \int_{-1}^{1} \left( \sin^{-1} x + \frac{x^5 + x^3 - 1}{\cos^2 x} \right) dx, \]

we will analyze the integrand to determine if it is odd or even.

### Step 1: Identify the functions in the integrand

The integrand consists of two parts: \( \sin^{-1} x \) and \( \frac{x^5 + x^3 - 1}{\cos^2 x} \).

### Step 2: Determine the nature of \( \sin^{-1} x \)

The function \( \sin^{-1} x \) is an odd function because:

\[ \sin^{-1}(-x) = -\sin^{-1}(x). \]

### Step 3: Analyze the second part \( \frac{x^5 + x^3 - 1}{\cos^2 x} \)

1. **Numerator**: The numerator \( x^5 + x^3 - 1 \) is an odd function because:
- \( x^5 \) is odd,
- \( x^3 \) is odd,
- \(-1\) is a constant (even).
Thus, the sum \( x^5 + x^3 - 1 \) is odd.

2. **Denominator**: The denominator \( \cos^2 x \) is an even function since \( \cos(-x) = \cos(x) \).

3. **Overall function**: The fraction \( \frac{x^5 + x^3 - 1}{\cos^2 x} \) is odd because it is the ratio of an odd function to an even function.

### Step 4: Combine the results

Since both \( \sin^{-1} x \) and \( \frac{x^5 + x^3 - 1}{\cos^2 x} \) are odd functions, their sum is also an odd function:

\[ f(x) = \sin^{-1} x + \frac{x^5 + x^3 - 1}{\cos^2 x} \text{ is odd.} \]

### Step 5: Evaluate the integral

Using the property of integrals of odd functions over symmetric limits:

\[ \int_{-a}^{a} f(x) \, dx = 0 \text{ for odd } f(x). \]

Thus, we have:

\[ \int_{-1}^{1} f(x) \, dx = 0. \]

### Conclusion

The value of the integral is

\[ \boxed{0}. \]