Class 12 Integrals Mock Test (CBSE)

Topics in Integrals

• Integration as inverse of differentiation
• Integration by substitution
• Integrals of some particular functions
• Integration by partial fractions
• Integration by parts
• Definite integrals
• Fundamental theorem of calculus
• Evaluation of definite integrals by substitution
• Properties of definite integrals

Integrals practice questions with answers

1. Q1Multiple choice·1 mark
   Evaluate $\int \frac{e^{\tan^{-1}x}}{1 + x^2} \, dx$.

   1. (a)$e^{\tan^{-1}x} + C$
   2. (b)$e^{x} + C$
   3. (c)$\frac{e^{\tan^{-1}x}}{1 + x^2} + C$
   4. (d)$\log(1 + x^2) + C$
   Show answer and step marks

   1. Correct option: (a) $e^{\tan^{-1}x} + C$+1

   Answer: (a)
2. Q2Multiple choice·1 mark
   The value of $\int \frac{dx}{x^2 - 4x + 8}$ is:

   1. (a)$\frac{1}{2} \log \left| \frac{x-4}{x} \right| + C$
   2. (b)$\frac{1}{4} \tan^{-1}\left(\frac{x-2}{2}\right) + C$
   3. (c)$\frac{1}{2} \tan^{-1}\left(\frac{x-2}{2}\right) + C$
   4. (d)$\tan^{-1}\left(\frac{x-2}{2}\right) + C$
   Show answer and step marks

   1. Correct option: (c). Completing the square, $x^2 - 4x + 8 = (x-2)^2 + 2^2$. Using $\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C$, we obtain $\frac{1}{2} \tan^{-1}\left(\frac{x-2}{2}\right) + C$.+1

   Answer: (c)
3. Q3Multiple choice·1 mark
   The value of $\int x e^{x^2} \, dx$ is:

   1. (a)$x e^{x^2} + C$
   2. (b)$\frac{1}{2} e^{x^2} + C$
   3. (c)$e^{x^2} + C$
   4. (d)$2e^{x^2} + C$
   Show answer and step marks

   1. Correct option: (b). Let $t = x^2$, then $dt = 2x \, dx$, so $x \, dx = \frac{1}{2} dt$. The integral becomes $\frac{1}{2} \int e^t \, dt = \frac{1}{2} e^t + C = \frac{1}{2} e^{x^2} + C$.+1

   Answer: (b)
4. Q4Multiple choice·1 mark
   Evaluate $\int \frac{1}{x^2 - 9} \, dx$.

   1. (a)$\frac{1}{6} \log\left|\frac{x-3}{x+3}\right| + C$
   2. (b)$\frac{1}{3} \log\left|\frac{x+3}{x-3}\right| + C$
   3. (c)$\frac{1}{6} \log\left|\frac{x+3}{x-3}\right| + C$
   4. (d)$\frac{1}{3} \log\left|\frac{x-3}{x+3}\right| + C$
   Show answer and step marks

   1. Correct option: (a) $\frac{1}{6} \log\left|\frac{x-3}{x+3}\right| + C$+1

   Answer: (a)

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