Expand the function \( \frac{1}{6+7 x} \) in a power series \( \sum_{n=0}^{\infty} a_{n} x^{n} \) with center \( c=0 \). Find \( a_{n} x^{n} \).

\( \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 2 x } f ( x , y ) d y d x \)

Evaluate the integrals i) \( \int_{2}^{4} \int_{0}^{x} \int_{0}^{x+y} z d x d y d x \)

A function \( f \) is defined as \( f(x)=\frac{5}{(x+2)^{2}}-\frac{2}{x+2} \) for \( x>0 \). Find an expression for \( f^{\prime}(x) \) and determine whether \( f \) is an increasing function, a decreasing function or aeither.

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

\( y ^ { \prime \prime } - ( \lambda _ { 1 } + \lambda _ { 2 } ) y ^ { \prime } + \lambda _ { 1 } \lambda _ { 2 } y = 0 , y = C _ { 1 } e ^ { \lambda _ { 1 } x } + C _ { 2 } e ^ { \lambda _ { 2 } x } \)