Question

\[\int\left( 3x\sqrt{x} + 4\sqrt{x} + 5 \right)dx\]

Answer 1

\[\int\left( 3x\sqrt{x} + 4\sqrt{x} + 5 \right)dx\]
\[ = \int\left( 3 x^1 \cdot x^\frac{1}{2} + 4 x^\frac{1}{2} + 5 \right)dx\]
\[ = 3\int x^\frac{3}{2} dx + 4\int x^\frac{1}{2} dx + 5 ∫dx\]
\[ = 3\left[ \frac{x^\frac{3}{2} + 1}{\frac{3}{2} + 1} \right] + 4\left[ \frac{x^\frac{1}{2} + 1}{\frac{1}{2} + 1} \right] + 5x + C\]
\[ = 3 \times \frac{2}{5} x^\frac{5}{2} + 4 \times \frac{2}{3} x^\frac{3}{2} + 5x + C\]
\[ = \frac{6}{5} x^\frac{5}{2} + \frac{8}{3} x^\frac{3}{2} + 5x + C\]