The simplified value of \(\rm \displaystyle\int x^2 \cos ax \ dx\)

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  1. \(\rm \frac{1}{a^3} (2ax \cos ax - 2 \sin ax + a^2 x^2 \sin ax)\)
  2. \(\rm \frac{1}{a^2} (2ax \cos ax - 2 \sin ax + a^2 x \sin ax)\)
  3. \(\rm \frac{1}{a} (2ax \cos ax - 2 \sin ax + a x \sin ax)\)
  4. \(\rm \frac{1}{a}(2ax \cos ax - 2 \sin ax + a^2 x^2 \sin ax)\)

Answer (Detailed Solution Below)

Option 1 : \(\rm \frac{1}{a^3} (2ax \cos ax - 2 \sin ax + a^2 x^2 \sin ax)\)
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Detailed Solution

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Concept:

The product of function can be integrated by the method of "Integration by parts".

By the method of integration by parts we have,

\(\smallint f\left( x \right)g\left( x \right)dx\; = \;f\left( x \right)\smallint g\left( x \right)dx - \smallint \left[ {f'\left( x \right)\smallint g\left( x \right)dx} \right]dx\)

Where f is the first function and g is the second function.

Which is chosen based on the order for the selection of the first function: ILATE (Inverse, Logarithmic, Algebraic, Trigonometric, Exponent)

Calculation:

We have, I = \(\rm \displaystyle∫ x^2 \cos ax \ dx\)

By using the above concept,

I = \(X^2\int cosaxdx-\int[2x\int cosaxdx]dx\)

or, I = \(\frac{X^2sinax}{a}-\int \frac{2xsinax}{a}dx\)

or, I = \(\frac{X^2sinax}{a}-\frac{2}{a}[x\int sinaxdx-\int[1\int sinaxdx]]dx\)

or, I = \(\frac{X^2sinax}{a}-\frac{2}{a}[-\frac{xcosax}{a}+\frac{sinax}{a^2}]\)

After solving it,

I = \(\frac{1}{a^3}[a^2x^2sinax+2axcosax-2sinax]\)

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