Questions: Evaluate the definite integral [ int8^8(9 x^10+1)^6 / 2 d x ]

Evaluate the definite integral [ int8^8(9 x^10+1)^6 / 2 d x ]
Transcript text: Evaluate the definite integral \[ \int_{8}^{8}\left(9 x^{10}+1\right)^{6 / 2} d x \]
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Solution

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Solution Steps

Step 1: Analyze the Integral

The integral to evaluate is: \[ \int_{8}^{8}\left(9 x^{10}+1\right)^{6 / 2} \, dx \] Notice that the lower and upper limits of integration are the same (both are 8). This means the integral is evaluated over a single point.

Step 2: Simplify the Integrand

Simplify the integrand: \[ \left(9 x^{10} + 1\right)^{6 / 2} = \left(9 x^{10} + 1\right)^3 \] However, this simplification is not necessary for solving the problem, as the integral is over a single point.

Step 3: Evaluate the Integral

When the lower and upper limits of integration are the same, the definite integral evaluates to 0. This is because the integral represents the area under the curve, and over a single point, the area is 0.

Final Answer

\[ \boxed{0} \]

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