The integral \int \frac{2x^{12}+5x^{9}}{\left ( x^{5}+x^{3}+1 \right )^{3}}dx is equal to:

  • Option 1)

    \frac{-x^{5}}{\left ( x^{5}+x^{3} +1\right )^{2}}+C

  • Option 2)

    \frac{x^{10}}{2\left ( x^{5}+x^{3} +1\right )^{2}}+C

  • Option 3)

    \frac{x^{5}}{2\left ( x^{5}+x^{3} +1\right )^{2}}+C

  • Option 4)

    \frac{-x^{10}}{2\left ( x^{5}+x^{3} +1\right )^{2}}+C

     

     

    where C is an arbitrary constant.

 

Answers (1)

As learnt in concept

Integration by substitution -

The functions when on substitution of the variable of integration to some quantity gives any one of standard formulas.

 

 

- wherein

Since \int f(x)dx=\int f(t)dt=\int f(\theta )d\theta all variables must be converted into single variable ,\left ( t\, or\ \theta \right )

 

 

 \int\frac{ 2x^{12}+5x^{9}}{\left ( x^{5}+x^{3}+1 \right )^{3}}dx

=\int \frac{\left ( \frac{2}{x^{3}}+\frac{5}{x^{6}} \right )}{\left ( 1+\frac{1}{x^{2}} +\frac{1}{x^{5}}\right )^{3}}dx

Put \:1+\frac{1}{x^{2}}+\frac{1}{x^{5}}=t

Differentiating, \left ( \frac{-2}{x^{3}}-\frac{5}{x^{6}} \right )dx=dt

-\left ( \frac{2}{x^{3}}+\frac{5}{x^{6}} \right )dx=dt

=>\int \frac{-dt}{t^{3}}=\frac{t^{-2}}{2}+C

=\frac{x^{10}}{2\left ( x^{5}+x^{3}+1 \right )^{2}}+C


Option 1)

\frac{-x^{5}}{\left ( x^{5}+x^{3} +1\right )^{2}}+C

Incorrect option   

Option 2)

\frac{x^{10}}{2\left ( x^{5}+x^{3} +1\right )^{2}}+C

Correct option

Option 3)

\frac{x^{5}}{2\left ( x^{5}+x^{3} +1\right )^{2}}+C

Incorrect option   

Option 4)

\frac{-x^{10}}{2\left ( x^{5}+x^{3} +1\right )^{2}}+C

 

 

where C is an arbitrary constant.

Incorrect option   

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