Questions: 已知函数 f(x)=∫1^x sqrt(1+t^3) dt, 则 ∫0^1 x f(x) dx=

Transcript text: 18. 已知函数 $f(x)=\int_{1}^{x} \sqrt{1+t^{3}} \mathrm{~d} t$, 则 $\int_{0}^{1} x f(x) \mathrm{d} x=$

Solution

Solution Steps

To solve the given integral, we need to use the definition of the function $ f(x) $ and then substitute it into the integral. We will use numerical integration to evaluate the integral.

Step 1: Define the Function $ f(x) $

Given the function: \[ f(x) = \int_{1}^{x} \sqrt{1 + t^3} \, dt \]

Step 2: Define the Integrand

We need to evaluate the integral: \[ \int_{0}^{1} x f(x) \, dx \]

Step 3: Evaluate the Integral

Using numerical integration, we find: \[ \int_{0}^{1} x f(x) \, dx \approx -0.2032 \]

Final Answer

$\boxed{-0.2032}$

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