If `int (x^(2020)+x^(804)+x^(402))(2x^(1608)+5x^(402)+10)^(1//402)dx=(1)/(10a)(2x^(2010)+5x^(804)+10^(402))^(a//402)`. Then `(a-400)` is equal to .......

Answer

To solve the given integral problem, we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int (x^{2020} + x^{804} + x^{402})(2x^{1608} + 5x^{402} + 10)^{\frac{1}{402}} \, dx \]

### Step 2: Factor Out Common Terms
First, we can factor out \(x\) from the first part of the integral:
\[ = \int x (x^{2019} + x^{803} + x^{401})(2x^{1608} + 5x^{402} + 10)^{\frac{1}{402}} \, dx \]

### Step 3: Substitute for Simplification
Let:
\[ t = 2x^{2010} + 5x^{804} + 10x^{402} \]
Now differentiate \(t\):
\[ dt = (2 \cdot 2010 x^{2009} + 5 \cdot 804 x^{803} + 10 \cdot 402 x^{401}) \, dx \]
This simplifies to:
\[ dt = (4020 x^{2009} + 4020 x^{803} + 4020 x^{401}) \, dx \]

### Step 4: Express \(dx\) in Terms of \(dt\)
From the above, we can express \(dx\):
\[ dx = \frac{dt}{4020 (x^{2009} + x^{803} + x^{401})} \]

### Step 5: Substitute Back into the Integral
Substituting \(t\) and \(dx\) into the integral gives:
\[ \int x \left( \frac{dt}{4020 (x^{2009} + x^{803} + x^{401})} \right) t^{\frac{1}{402}} \]

### Step 6: Simplify the Integral
This can be simplified to:
\[ \frac{1}{4020} \int t^{\frac{1}{402}} \, dt \]

### Step 7: Integrate
Using the power rule for integration:
\[ \int t^{n} \, dt = \frac{t^{n+1}}{n+1} + C \]
we have:
\[ \int t^{\frac{1}{402}} \, dt = \frac{t^{\frac{1}{402} + 1}}{\frac{1}{402} + 1} + C = \frac{t^{\frac{403}{402}}}{\frac{403}{402}} + C = \frac{402}{403} t^{\frac{403}{402}} + C \]

### Step 8: Substitute Back for \(t\)
Substituting back for \(t\):
\[ = \frac{1}{4020} \cdot \frac{402}{403} \left(2x^{2010} + 5x^{804} + 10x^{402}\right)^{\frac{403}{402}} + C \]

### Step 9: Compare with Given Expression
We compare this with the given expression:
\[ \frac{1}{10a} \left(2x^{2010} + 5x^{804} + 10x^{402}\right)^{\frac{a}{402}} \]
From the comparison:
\[ \frac{1}{4020} \cdot \frac{402}{403} = \frac{1}{10a} \]
This leads to:
\[ \frac{1}{4030} = \frac{1}{10a} \]

### Step 10: Solve for \(a\)
Cross-multiplying gives:
\[ 10a = 4030 \implies a = 403 \]

### Step 11: Find \(a - 400\)
Finally, we calculate:
\[ a - 400 = 403 - 400 = 3 \]

### Final Answer
Thus, the value of \(a - 400\) is:
\[ \boxed{3} \]