Find the antiderivative for the given function. \( \int \frac{7 d x}{\sqrt{49 x^{2}-1}}=\square+C \) Note: Enter inverse functions using \( { }^{\wedge 1} \) notation with no parentheses a For example, use \( \sinh ^{\wedge} 1(\mathrm{x}) \) for \( \sinh ^{-1}(x) \).

Evaluate the expression \( \int_{3}^{5} x^{2} \sin \sqrt{x} \ln x d x+\int_{5}^{3} x^{2} \sin \sqrt{x} \ln x d x \) Answer: \( \square \)

Use one or more definite integrals to find the area between the \( x \)-axis and the given curve on the given interval. Be sure to find if the curve crosses the \( x \)-axis in the given interval. \( f(x)=4 e^{x}-4 \) on \( [-2,2] \) Area:

Evaluate the definite integral \( \int_{-\pi / 2}^{\pi / 2} 8 \cos x d x \) Answer:

Use one or more definite integrals to find the area between the \( x \)-axis and the given curve on the given interval. Be sure to find if the curve crosses the \( x \)-axis in the given interval. \( f(x)=3-3 x^{2} \) on \( [0,2] \) Area:

\( \iint_{R} d A \), si \( R \) es la región, dada por: \[ R=\left\{(x, y) \in \mathbb{R}^{2}, 0 \leq x \leq \frac{1}{2}, y+x \leq 1, y \geq 0 .\right\} \]