Evalutate `intx^(3)sinx^(4)dx.`

Answer

To evaluate the integral \( \int x^3 \sin(x^4) \, dx \), we can use the substitution method. Here are the steps to solve the integral:

### Step 1: Choose a substitution
Let \( t = x^4 \). Then, we need to find \( dt \):
\[ dt = 4x^3 \, dx \quad \Rightarrow \quad dx = \frac{dt}{4x^3} \]

### Step 2: Substitute in the integral
Now, substitute \( t \) and \( dx \) into the integral:
\[ \int x^3 \sin(x^4) \, dx = \int x^3 \sin(t) \cdot \frac{dt}{4x^3} \]
The \( x^3 \) terms cancel out:
\[ = \frac{1}{4} \int \sin(t) \, dt \]

### Step 3: Integrate
Now, we can integrate \( \sin(t) \):
\[ \int \sin(t) \, dt = -\cos(t) + C \]
Thus,
\[ \frac{1}{4} \int \sin(t) \, dt = -\frac{1}{4} \cos(t) + C \]

### Step 4: Substitute back
Now, substitute back \( t = x^4 \):
\[ -\frac{1}{4} \cos(x^4) + C \]

### Final answer
The final result of the integral is:
\[ \int x^3 \sin(x^4) \, dx = -\frac{1}{4} \cos(x^4) + C \]