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Indian Navy Pilot Definite Integration — practice questions

Explore 23 free Definite Integration questions in Indian Navy Pilot Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Definite Integration questions and exercises with answers

Showing questions 1–23 of 23.

If $f(x) = x^3 \sin^4 x$, then $\int_{-\pi/4}^{\pi/4} f(x)\,dx$ equals:Read the answer and worked solution →$\int_0^{\pi/2} \sqrt{1 - \cos 4x}\,dx$ equals:Read the answer and worked solution →Order the steps to integrate ∫ x·eˣ dx by partsRead the answer and worked solution →Complete the relationship between the two calculiRead the answer and worked solution →True or false: the constant of integrationRead the answer and worked solution →Differentiation or integration?Read the answer and worked solution →Tap every property of indefinite integrals the chapter givesRead the answer and worked solution →Properties in useRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →Order the steps: check an indefinite integralRead the answer and worked solution →Flip and pair: term with meaningRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →True or false: reading the notationRead the answer and worked solution →Definite or indefinite?Read the answer and worked solution →Reading the definitions backRead the answer and worked solution →Complete the linearity rulesRead the answer and worked solution →Which process is being described?Read the answer and worked solution →True or false: what the constant meansRead the answer and worked solution →Checking an integralRead the answer and worked solution →Tap every true statement about integrationRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →

Worked practice questions

If $f(x) = x^3 \sin^4 x$, then $\int_{-\pi/4}^{\pi/4} f(x)\,dx$ equals:

  1. $0$
  2. $\pi/4$
  3. $1$
  4. $2\int_0^{\pi/4} f\,dx$
Show answer and solution

Correct answer: A. $0$

$f(-x) = (-x)^3 \sin^4(-x) = -x^3 \cdot \sin^4 x = -f(x)$. So $f$ is **odd**, and over the symmetric interval the integral is 0.

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$\int_0^{\pi/2} \sqrt{1 - \cos 4x}\,dx$ equals:

  1. $\sqrt 2$
  2. $1$
  3. $2$
  4. $\pi$
Show answer and solution

Correct answer: A. $\sqrt 2$

Use $1 - \cos 4x = 2\sin^2 2x$, so $\sqrt{1-\cos 4x} = \sqrt 2|\sin 2x|$. On $[0, \pi/2]$, $\sin 2x \ge 0$ (since $2x \in [0,\pi]$). $\int_0^{\pi/2} \sqrt 2 \sin 2x\,dx = \sqrt 2 [-\cos 2x/2]_0^{\pi/2} = \sqrt 2 [1/2 + 1/2] = \sqrt 2$.

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Order the steps to integrate ∫ x·eˣ dx by parts

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Steps
  1. Identify u = x and dv = eˣ dx (using ILATE)
  2. Compute du = dx and v = eˣ
  3. Apply ∫u dv = uv − ∫v du
  4. Substitute: x·eˣ − ∫eˣ dx
  5. Evaluate the remaining integral: x·eˣ − eˣ + C

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