Determine a integral de \( \text{\int} (5x^4 + 3x^2)^{10} \times (20x^3 + 6x) \ dx \ ) utilizando o método de substituição.

The series \( \sum_{\substack{k=1}}^{\infty} \frac{(-1)^{k}}{\sqrt{k}} \) converges. True False

\( f(x)=x^{3}-3 x^{2}+3 x+5 \) Consider the function \( f(x, y)=\sqrt{x+y^{2}} \). Calculate \( f_{x y}=\frac{\partial^{2} f}{\partial x \partial y} \)

\( \int \frac{e^{3 \mathrm{x}} \mathrm{dx}}{\sqrt{\mathrm{e}^{3 \mathrm{x}}+2}} \) (use substitution: \( u=e^{3 \mathrm{x}}+2^{\text {) }} \)

\( \int 4 e^{-6 x} d x \) (use substitution: \( u=-6 x \) )

6 A curve is such that \( \frac{d y}{d x}=\frac{4}{\sqrt{2 x+k}} \), where \( k \) lo a constant. The point \( P(3,2) \) lies on the curve and tho norma! to the curve at \( P \) is \( x+4 y=11 \). Find the equation of the eurve.