The integral ( x 8 - x 2 ) d x ( x 12 + 3 x 6 + 1 ) tan - 1 ( x 3 + 1 x 3 ) is equal to: [2024]

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Q.
The integral $\int \frac{(x^{8} - x^{2}) d x}{(x^{12} + 3 x^{6} + 1) \tan^{- 1} (x^{3} + \frac{1}{x^{3}})}$ is equal to: [2024]

1 $\log_{e}$ $(| \tan^{- 1} (x^{3} + \frac{1}{x^{3}}) |) + C$

2 $\log_{e}$ $(| \tan^{- 1} {(x^{3} + \frac{1}{x^{3}}) |)}^{3} + C$

3 $\log_{e}$ $(| \tan^{- 1} {(x^{3} + \frac{1}{x^{3}}) |)}^{\frac{1}{3}} + C$

4 $\log_{e}$ $(| \tan^{- 1} {(x^{3} + \frac{1}{x^{3}}) |)}^{\frac{1}{2}} + C$

Ans.
(3)

Let $I = \int \frac{(x^{8} - x^{2}) d x}{(x^{12} + 3 x^{6} + 1) \tan^{- 1} (x^{3} + \frac{1}{x^{3}})}$

Put $\tan^{- 1} (x^{3} + \frac{1}{x^{3}}) = u$

$\Rightarrow \frac{1}{1 + {(x^{3} + \frac{1}{x^{3}})}^{2}} \cdot 3 (x^{2} - \frac{1}{x^{4}}) d x = d u$

$\Rightarrow (\frac{x^{6}}{x^{12} + 3 x^{6} + 1} \times \frac{x^{6} - 1}{x^{4}}) d x = \frac{d u}{3}$

$\Rightarrow I = \frac{1}{3} \int \frac{d u}{u} = \frac{\ln | u |}{3} = \ln$ $(| \tan^{- 1} {(x^{3} + \frac{1}{x^{3}}) |)}^{\frac{1}{3}} + C$

Solution

Q.
The integral $\int \frac{(x^{8} - x^{2}) d x}{(x^{12} + 3 x^{6} + 1) \tan^{- 1} (x^{3} + \frac{1}{x^{3}})}$ is equal to: [2024]

1 $\log_{e}$ $(| \tan^{- 1} (x^{3} + \frac{1}{x^{3}}) |) + C$

2 $\log_{e}$ $(| \tan^{- 1} {(x^{3} + \frac{1}{x^{3}}) |)}^{3} + C$

3 $\log_{e}$ $(| \tan^{- 1} {(x^{3} + \frac{1}{x^{3}}) |)}^{\frac{1}{3}} + C$

4 $\log_{e}$ $(| \tan^{- 1} {(x^{3} + \frac{1}{x^{3}}) |)}^{\frac{1}{2}} + C$

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Solution

Q. The integral ∫(x8-x2)dx(x12+3x6+1)tan-1(x3+1x3) is equal to: [2024]

1 loge(|tan-1(x3+1x3)|)+C

2 loge(|tan-1(x3+1x3)|)3+C

3 loge(|tan-1(x3+1x3)|)13+C

4 loge(|tan-1(x3+1x3)|)12+C

Ans. (3) Let I=∫(x8-x2) dx(x12+3x6+1)tan-1(x3+1x3) Put tan-1(x3+1x3)=u ⇒11+(x3+1x3)2·3(x2-1x4)dx=du ⇒(x6x12+3x6+1×x6-1x4)dx=du3 ⇒I=13∫duu=ln|u|3=ln(|tan-1(x3+1x3)|)13+C

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Q.
The integral $\int \frac{(x^{8} - x^{2}) d x}{(x^{12} + 3 x^{6} + 1) \tan^{- 1} (x^{3} + \frac{1}{x^{3}})}$ is equal to: [2024]

1 $\log_{e}$ $(| \tan^{- 1} (x^{3} + \frac{1}{x^{3}}) |) + C$

2 $\log_{e}$ $(| \tan^{- 1} {(x^{3} + \frac{1}{x^{3}}) |)}^{3} + C$

3 $\log_{e}$ $(| \tan^{- 1} {(x^{3} + \frac{1}{x^{3}}) |)}^{\frac{1}{3}} + C$

4 $\log_{e}$ $(| \tan^{- 1} {(x^{3} + \frac{1}{x^{3}}) |)}^{\frac{1}{2}} + C$