What is the integral of f(x) = 1 + x² + x⁴ with respect to x²?

Question

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

Concept:

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

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

Calculation:

Let, x² = u

From equation (i)

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

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

Now putting the value of u,

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

∴ The required integral is x2 + \(\rm \frac{x^{4}}{2}\) + \(\rm \frac{x^{6}}{3}\) + C.