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Practice Problems for 4/29

  1. Suppose \(F(x)=\int_0^x \frac{2}{t^3+1} dt\). Explain why \(F\) is differentiable at \(x=1\), then find the value of \(F'(1)\).

  2. Compute \(\frac{d}{dx}\int_x^{\sin x} 3t^2\, dt\).

  3. Which of the following can be evaluated using the Fundamental Theorem of Calculus Part 2?
    1. \( \int_1^5 \frac{x-3}{x^2+1}\,dx\)

    2. \( \int_1^5 \frac{x-3}{x^2-4}\, dx\)

    3. \(\int_{-1}^1 \frac{1}{x}\, dx\)

    4. \(\int_{1/2}^1 \frac{1}{x}\, dx\)

    5. \( \int_{-1}^1 \sqrt{1-x^2}\,dx\)


  4. Evaluate \( \int_0^1 4x^3-9x^2+2x-3\, dx\)