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

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 $\int_{0}^{1} x f(x) \mathrm{d} x=$
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Solution

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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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