Q.Find the integral:
\int \left(\sqrt{x} - \frac{1}{\sqrt{x}}\right)^2 dx

Split the integrand into two standard integrals, $\sec^2x$ and $\sec x\tan x$.

Working: Split the fraction:

$\int \frac{2-3\sin x}{\cos^2 x}\,dx = \int \frac{2}{\cos^2 x}\,dx - \int\frac{3\sin x}{\cos^2 x}\,dx = 2\int\sec^2x\,dx - 3\int \sec x\tan x\,dx$

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$\displaystyle\int\left(\sqrt x-\tfrac{1}{\sqrt x}\right)^2 dx=\dfrac{x^2}{2}-2x+\ln|x|+c$.

Concept. Expand the integrand into simple power terms, then integrate term by term using $\int x^n dx=\dfrac{x^{n+1}}{n+1}$ and $\int \dfrac1x dx=\ln|x|$.

Steps. …

Split the product and integrate each standard term separately.

$\int \csc x(\csc x+\cot x)\,dx = \int \csc^2 x\,dx + \int \csc x\cot x\,dx$

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$\displaystyle\int x(4x^{2}-6)\,dx=x^{4}-3x^{2}+k$.

Expand the integrand: $x(4x^{2}-6)=4x^{3}-6x$. Integrate term by term:

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Simplify the integrand to a polynomial, then integrate term by term.

$\int x^2\left(1-\frac1{x^2}\right)dx=\int\left(x^2-1\right)dx=\frac{x^3}{3}-x+c.$

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Expand the integrand into standard forms whose antiderivatives are known.

$\int \sec x(\sec x + \tan x)\,dx = \int (\sec^2 x + \sec x\tan x)\,dx.$

Using $\int \sec^2 x\,dx = \tan x$ and $\int \sec x\tan x\,dx = \sec x$,

$= \tan x + \sec x + C.$

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Write $(1-x)\sqrt x = x^{1/2}-x^{3/2}$ and integrate: $\dfrac23x^{3/2}-\dfrac25x^{5/2}+C$.

Concept. Power rule: $\displaystyle\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C$ (for $n e-1$).

Working.

$(1-x)\sqrt x = \sqrt x - x\sqrt x = x^{1/2}-x^{3/2}.$ …

$\displaystyle\int\dfrac{x^3-1}{x^2}\,dx=\dfrac{x^2}{2}+\dfrac{1}{x}+C$.

Concept. Simplify a rational integrand by term-by-term division, then use the power rule $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C$.

Step-by-step.

$\frac{x^3-1}{x^2}=\frac{x^3}{x^2}-\frac{1}{x^2}=x-x^{-2}.$ …