x²के संबंध में f(x) = 1 + x² + x⁴का समाकलन क्या है?

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Question

This question was previously asked in

NDA 02/2021: Maths Previous Year paper (Held On 14 Nov 2021)

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Answer 1

अवधारणा:

\(\rm \int x^{n}dx = \frac{x^{n + 1}}{n + 1} + C\)

\(\rm \int f(x)dx^2\) = \(\rm \int (1 + x^{2} + x^{4})d(x^2)\) .....(i)

गणना:

माना, x2 = u

समीकरण (i) से

\(\rm \int f(x)dx^2\) = \(\rm \int (1 + u + u^{2})du\)

= u + \(\rm \frac{u^{2}}{2}\) + \(\rm \frac{u^{3}}{3}\) + C

अब u का मान रखते हुए,

⇒ \(\rm \int f(x)dx^2\) = x2 + \(\rm \frac{x^{4}}{2}\) + \(\rm \frac{x^{6}}{3}\) + C

∴ आवश्यक समाकलन x2 + \(\rm \frac{x^{4}}{2}\) + \(\rm \frac{x^{6}}{3}\) + C है