Question

Find the following integral : $\int \biggr\lgroup{\sqrt{x}-\frac{1}{\sqrt{x}}\biggr\rgroup^2 \, dx$

Answer 1

Answer:

Step-by-step explanation:

Concept:

The given integral is solved by decomposition method.

In decomposition method, the given integrand(non-integrable function) is decomposed into integrable functions by using algebraic identities, trigonometric identities, etc.

$(a-b)^2=a^2+b^2-2ab$

Now,

$\int{(\sqrt{x}-\frac{1}{\sqrt{x}})^2}\:dx\\\\=\int[(\sqrt{x})^2+(\frac{1}{\sqrt{x}})^2-2.\sqrt{x}.\frac{1}{\sqrt{x}}]\:dx\\\\=\int[x+\frac{1}{x}-2]\:dx\\\\=\int{x}\:dx+\int{\frac{1}{x}}\:dx-2\int\:dx\\\\=\frac{x^2}{2}+logx-2x+c$