`int x sinx sec^(3)x dx` is equal to

Options

A. `(1)/(2)[ sec^(2)x-tanx]+c`
B. `(1)/(2)[x sec^(2)x-tanx]+c`
C. `(1)/(2)[x sec^(2)x+tanx]+c`
D. `(1)/(2)[ sec^(2)x+tanx]+c`

Explanation

To solve the integral \( \int x \sin x \sec^3 x \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting \( \sec^3 x \) in terms of sine and cosine:
\[ \sec^3 x = \frac{1}{\cos^3 x} \]
Thus, the integral becomes:
\[ \int x \sin x \sec^3 x \, dx = \int x \sin x \frac{1}{\cos^3 x} \, dx \]

### Step 2: Use Trigonometric Identity
Using the identity \( \sin x = \tan x \cos x \), we can rewrite the integral:
\[ \int x \sin x \sec^3 x \, dx = \int x \tan x \sec^2 x \, dx \]

### Step 3: Integration by Parts
Now we will apply integration by parts. Let:
- \( u = x \) (thus \( du = dx \))
- \( dv = \tan x \sec^2 x \, dx \) (thus \( v = \sec x \))

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \):
\[ \int x \tan x \sec^2 x \, dx = x \sec x - \int \sec x \, dx \]

### Step 4: Integrate \( \sec x \)
The integral of \( \sec x \) is:
\[ \int \sec x \, dx = \ln |\sec x + \tan x| + C \]
Thus, substituting back, we have:
\[ \int x \tan x \sec^2 x \, dx = x \sec x - \ln |\sec x + \tan x| + C \]

### Step 5: Final Result
Putting everything together, we find:
\[ \int x \sin x \sec^3 x \, dx = x \sec x - \ln |\sec x + \tan x| + C \]