Question

0,64


2,36


1,64


1,31


0,86

Answer 1

$\int_{0}^{1}\int_{0}^{x} (\sqrt{4x^2+5})\;dy\;dx \\\\ \int_{0}^{1} \left (y\sqrt{4x^2+5}\ \right |^x_0)\;dx \\\\\ \int_{0}^{1} (x\sqrt{4x^2+5})\;dx \\\\\text{substituicao}\\\\ u=4x^2+5\\du=8x\;dx\\\\ \frac{du}{8x} =dx\\\\ \text{substituindo na integral}\\\\ \int_0^1 \not x \sqrt{u} *\; \frac{du}{8\not x} \\\\ \frac{1}{8}\int_0^1 u^{ \frac{1}{2} } \;du = \frac{1}{8}\left( \frac{u\frac{3}{2}}{\frac{3}{2}} \right)^1_0 = \frac{1}{8}*\left( \frac{2u^\frac{3}{2}}{3} \right)^1_0$

$= \frac{1}{8}*\left( \frac{2(4x^2+5)^\frac{3}{2}}{3}\right ) ^1_0 = \frac{1}{8}*\left( \frac{2(4+5)^\frac{3}{2}}{3}- \frac{2(0+5)^\frac{3}{2}}{3}\right ) \approx 1,318$