Integrate the function- - 1 -tan -1 x d x

Let I =∫ 1.tan^ 1x dx ⇒ I =tan^ 1x ∫ 1 dx ∫ [ d/dx(tan^ 1)∫ 1 dx ]dx =x tan^ 1x ∫1/1+x^2.x dx Let 1+x^2 =t ⇒ 2x=dt/dx⇒ dx =dt/2x ∴ I =x tan^ 1x ∫x/t.dt/2x=x ...

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Standard XII

Mathematics

Definite Integral as Limit of Sum

Integrate the...

Question

Integrate the function.
$\int 1. t a n^{- 1} x d x$

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Solution

Let $I = \int 1. t a n^{- 1} x d x$
$\Rightarrow I = t a n^{- 1} x \int 1 d x - \int [\frac{d}{d x} (t a n^{- 1}) \int 1 d x] d x = x t a n^{- 1} x - \int \frac{1}{1 + x^{2}} . x d x$
Let $1 + x^{2} = t \Rightarrow 2 x = \frac{d t}{d x} \Rightarrow d x = \frac{d t}{2 x}$
$∴ I = x t a n^{- 1} x - \int \frac{x}{t} . \frac{d t}{2 x} = x t a n^{- 1} x - \frac{1}{2} \int \frac{1}{t} d t = x t a n^{- 1} x - \frac{1}{2} l o g | t | + C = x t a n^{- 1} x - \frac{1}{2} l o g | 1 + x^{2} | + C$

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Integrate the function. ∫1.tan−1xdx

Let I=∫1.tan−1xdx ⇒I=tan−1x∫1dx−∫[ddx(tan−1)∫1dx]dx=xtan−1x−∫11+x2.xdx Let 1+x2=t⇒2x=dtdx⇒dx=dt2x ∴I=xtan−1x−∫xt.dt2x=xtan−1x−12∫1tdt=xtan−1x−12log|t|+C=xtan−1x−12log|1+x2|+C

Integrate the function.
$\int 1. t a n^{- 1} x d x$