What is the integral of f(x) = 1 + x² + x^{4} with respect to x²?

Correct Answer is : rmx2+x42+x63+C{rm} x² + x^4/2+x^6/3+Crmx2+2x4+3x6+C

{rm} x² + {x^{4}}{4}+{x^{6}}{6}+C

Correct Answer is : rmx2+x42+x63+C{rm} x² + x^4/2+x^6/3+Crmx2+2x4+3x6+C

Test Index

UPSC NDA 2 2021 Mathematics Solved Paper

Question : 81 of 120

Marks: +1, -0

What is the integral of

f(x) = 1 + x^{2} + x^{4}

x^{2}

Solution:

Concept:

The integral with respect to

x^2

means treating

x^2

as the variable of integration. Use the substitution

u = x^2

and the power rule

\int u^n \, du = \frac{u^{n+1}}{n+1} + C

Explanation:

We need to evaluate

\int (1 + x^2 + x^4) \, d(x^2)

Let

u = x^2

. Then

d(x^2) = du

Substitute:

\int (1 + x^2 + x^4) \, d(x^2) = \int (1 + u + u^2) \, du

Integrate term by term:

\int 1 \, du = u

\int u \, du = \frac{u^2}{2}

\int u^2 \, du = \frac{u^3}{3}

So the result is

u + \frac{u^2}{2} + \frac{u^3}{3} + C

Now replace

u

with

x^2

x^2 + \frac{(x^2)^2}{2} + \frac{(x^2)^3}{3} + C = x^2 + \frac{x^4}{2} + \frac{x^6}{3} + C

Thus the required integral is

x^2 + \frac{x^4}{2} + \frac{x^6}{3} + C

Answer:

x^2 + \frac{x^4}{2} + \frac{x^6}{3} + C

(Option D).

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