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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2021 · Mathematics · Indefinite Integration

What is the integral of f(x)=1+x^2+x^4 with respect to x^2 ?

What is the integral of f(x)=1+x2+x4f(x)=1+x^2+x^4 with respect to x2?x^2 ?

Options

  1. Ax+x33+x55+Cx+\frac{x^3}{3}+\frac{x^5}{5}+C
  2. Bx33+x55+C\frac{x^3}{3}+\frac{x^5}{5}+C
  3. Cx2+x44+x66+Cx^2+\frac{x^4}{4}+\frac{x^6}{6}+C
  4. Dx2+x42+x63+Cx^2+\frac{x^4}{2}+\frac{x^6}{3}+C

Correct answer

D. x2+x42+x63+Cx^2+\frac{x^4}{2}+\frac{x^6}{3}+C

Step-by-step solution

Let x2=tx^2=tf(x)=1+x2+x4I=∫f(x)dx2=∫(1+x2+x4)dx2=∫(1+t+t2)dt=t+t22+t33+cI=x2+x42+x63+C\begin{aligned} & f(x)=1+x^2+x^4 \\ & \mathrm{I}=\int f(x) d x^2=\int\left(1+x^2+x^4\right) d x^2 \\ & =\int\left(1+t+t^2\right) d t=t+\frac{t^2}{2}+\frac{t^3}{3}+c \\ & \mathrm{I}=x^2+\frac{x^4}{2}+\frac{x^6}{3}+C \end{aligned}
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