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BITSAT 2009

Exam: BITSAT
Year: 2009
Subject: Mathematics
Chapter: Definite Integration

BITSAT 2009 · Mathematics · Definite Integration

Evaluate: ( ^ -1 x d x)

Evaluate: (\intx \tan ^{-1} x d x)

Options

A( $\frac{1}{2}\left($ x^2+1\right) \tan ^{-1} x- $\frac{1}{2}$ x+c)
B( $\frac{1}{2}\left($ x^2+1\right) \tan ^{-1} x+ $\frac{1}{2}$ x+c)
C( $\frac{1}{2}\left(\mathrm{x}^$ 2-1\right) \tan ^{-1} $\mathrm{x}-\frac{1}{2} \mathrm{x}+\mathrm{c}$ )
DNone of these

Correct answer

A. ( $\frac{1}{2}\left($ x^2+1\right) \tan ^{-1} x- $\frac{1}{2}$ x+c)

Step-by-step solution

\begin{aligned}& \int x \tan ^{-1} x d x=\frac{x^2}{2} \tan ^{-1} x-\frac{1}{2} \int \frac{x^2}{1+x²}d x \\ & =\frac{x^2}{2} \tan ^{-1}x-\frac{1}{2} \int \lleft[1-\frac{1}{1+x²}]d x \\ & =\frac{1}{2}\left(x^2+1\right)\tan ^{-1}x-\frac{1}{2} x+c \end{aligned}

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