"Evaluate the following integrals. ∫012xtan−1(x2)dx" was answered by our community

floymdiT

Answered question

2021-10-11

Evaluate the following integrals.

\int_{0}^{1} 2 x \tan^{- 1} (x^{2}) d x

gotovub

Skilled2021-10-12Added 98 answers

Here,

\int_{0}^{1} 2 x \tan^{- 1} (x^{2}) d x

Let

t = x^{2}

. So, dt = 2xdx
Also, when

x = 0 \Rightarrow t = 0

and

x = 1 \Rightarrow t = 1

So,

\int_{0}^{1} 2 x \tan^{- 1} (x^{2}) d x

= \int_{0}^{1} \tan^{- 1} t d t

Step 2
By the method of by parts,

= \int_{0}^{1} \tan^{- 1} t d t

= {[t \tan^{- 1} t]}_{0}^{1} - \int_{0}^{1} t \frac{d}{d t} (\tan^{- 1} t) d t

= (\frac{π}{4} - 0) - \int_{0}^{1} t \times \frac{1}{1 + t^{2}} d t

= \frac{π}{4} - \int_{0}^{1} \frac{t}{1 + t^{2}} d t

= \frac{π}{4} - {[\frac{1}{2} \ln (1 + t^{2})]}_{0}^{1}

= \frac{π}{4} - (\frac{1}{2} \ln 2 - 0)

= \frac{π}{4} - \frac{1}{2} \ln 2

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