Question

∫(1+∜x)dx÷(1+√x) .please solve this problem

Answer 1

Answer on Question #50468 – Math – Integral Calculus

$\int (1 + \sqrt[4]{x}) dx \div (1 + \sqrt{x})$ . please solve this problem

Solution

Evaluate

$$\begin{array}{l} \int \left(1 + \sqrt[4]{x}\right) dx = |additive\ property| = \int dx + \int \sqrt[4]{x} dx = \left| \text{formula} \int x^{\alpha} dx = \frac{x^{\alpha+1}}{\alpha+1} + C \right| = \\ = x + \frac{\frac{1}{x^{4+1}}}{\frac{1}{4+1}} + C = x + \frac{x^{5/4}}{5/4} + C = x + \frac{4}{5} x \cdot \sqrt[4]{x} + C, \end{array}$$

where $C$ is an arbitrary real constant.

Finally get

$$\frac{\int (1 + \sqrt[4]{x}) dx}{1 + \sqrt{x}} = \frac{x + \frac{4}{5} x \cdot \sqrt[4]{x} + C}{1 + \sqrt{x}},$$

where $C$ is an arbitrary real constant.

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