The integral cannot be expressed in terms of elementary functions, that is, the answer is not a finite combination of radicals, exponentials, logarithms, trigon ...

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Standard XII

Mathematics

Integration of Irrational Algebraic Fractions - 2

Integrate xta...

Question

Integrate x tanx.

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Solution

Given $\int x tan x d x$
Using the rule of $\int u v d x = u \int v d x - \int [u^{'} \int v d x] d x$
Now, replace $u = x$ and $v = tan x$ , we get
$\Rightarrow \int x tan x d x = x \int tan x d x - \int [(x)^{'} \int tan x d x] d x$
$= x ln cos x + \int ln cos x d x$ [Since, $\int tan x = - ln cos x + c$ ]
$∴ \int x tan x d x = x ln | cos x | + cos x ln | cos x | - cos x + c$ [Since $\int ln | x | = x ln x - x + c$ ]

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Integrate x tanx.

Given ∫xtanxdxUsing the rule of ∫uvdx=u∫vdx−∫[u′∫vdx]dxNow, replace u=x and v=tanx, we get ⇒∫xtanxdx=x∫tanxdx−∫[(x)′∫tanxdx]dx=xlncosx+∫lncosxdx [Since, ∫tanx=−lncosx+c]∴∫xtanxdx=xln|cosx|+cosxln|cosx|−cosx+c [Since ∫ln|x|=xlnx−x+c]