`intx^(2)secx^(3)dx=`

Options

A. `log(secx^(3)+tanx^(3))+c`
B. `3(secx^(3)+tanx^(3))+c`
C. `1/3log(secx^(3)+tanx^(3))+c`
D. `log(secx+tanx)+c`

Answer

To solve the integral \( \int x^2 \sec(x^3) \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = x^3 \). Then, differentiate both sides to find \( dx \):
\[ dt = 3x^2 \, dx \implies dx = \frac{dt}{3x^2} \]

### Step 2: Rewrite the Integral
Substituting \( t \) into the integral gives:
\[ \int x^2 \sec(x^3) \, dx = \int x^2 \sec(t) \cdot \frac{dt}{3x^2} \]
The \( x^2 \) terms cancel out:
\[ = \frac{1}{3} \int \sec(t) \, dt \]

### Step 3: Integrate Secant
The integral of \( \sec(t) \) is a standard result:
\[ \int \sec(t) \, dt = \ln |\sec(t) + \tan(t)| + C \]
Thus, we have:
\[ \frac{1}{3} \int \sec(t) \, dt = \frac{1}{3} \left( \ln |\sec(t) + \tan(t)| + C \right) \]

### Step 4: Substitute Back
Now substitute \( t = x^3 \) back into the expression:
\[ = \frac{1}{3} \ln |\sec(x^3) + \tan(x^3)| + C \]

### Final Answer
Thus, the final solution is:
\[ \int x^2 \sec(x^3) \, dx = \frac{1}{3} \ln |\sec(x^3) + \tan(x^3)| + C \]

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