If `int x^(6) sin ( 5x^(7)) dx = ( k )/( 5) cos ( 5x^(7))+C`, then the value of k is

Options

A. `- ( 1)/( 7 )`
B. `( 1)/( 7 )`
C. `(1)/( 35)`
D. `- (1)/( 35)`

Answer

To solve the integral \( \int x^6 \sin(5x^7) \, dx \) and find the value of \( k \) in the expression \( \frac{k}{5} \cos(5x^7) + C \), we will use substitution and integration techniques. Here is the step-by-step solution:

### Step 1: Substitution
Let \( t = 5x^7 \). Then, we differentiate \( t \) with respect to \( x \):
\[ dt = 35x^6 \, dx \quad \Rightarrow \quad dx = \frac{dt}{35x^6} \]

### Step 2: Rewrite the Integral
Substituting \( t \) into the integral, we have:
\[ \int x^6 \sin(5x^7) \, dx = \int x^6 \sin(t) \cdot \frac{dt}{35x^6} \]
This simplifies to:
\[ \int \sin(t) \cdot \frac{dt}{35} = \frac{1}{35} \int \sin(t) \, dt \]

### Step 3: Integrate
The integral of \( \sin(t) \) is:
\[ \int \sin(t) \, dt = -\cos(t) + C \]
Thus, we have:
\[ \frac{1}{35} \int \sin(t) \, dt = \frac{1}{35} (-\cos(t)) + C = -\frac{1}{35} \cos(t) + C \]

### Step 4: Substitute Back
Now, we substitute back \( t = 5x^7 \):
\[ -\frac{1}{35} \cos(5x^7) + C \]

### Step 5: Compare with Given Expression
We know from the problem statement that:
\[ \int x^6 \sin(5x^7) \, dx = \frac{k}{5} \cos(5x^7) + C \]
Setting the two expressions equal gives:
\[ -\frac{1}{35} \cos(5x^7) + C = \frac{k}{5} \cos(5x^7) + C \]
By canceling \( C \) from both sides, we get:
\[ -\frac{1}{35} = \frac{k}{5} \]

### Step 6: Solve for \( k \)
To find \( k \), we multiply both sides by 5:
\[ k = -\frac{5}{35} = -\frac{1}{7} \]

Thus, the value of \( k \) is:
\[ \boxed{-\frac{1}{7}} \]