alleryStart learning →∫₀^(π) sin² x dx equals:
- π
- π/2
- 2π
- π/4
Show answer and solution
Correct answer: B. π/2
sin² x = (1 - cos 2x)/2. So ∫₀^π sin² x dx = (1/2)∫₀^π (1 - cos 2x) dx = (1/2)[x - sin(2x)/2]₀^π = (1/2)[π - 0] = π/2.
Indian Navy INET Integrals — practice questions
Explore 24 free Integrals questions in Indian Navy INET Mathematics. Open each question or exercise to read its content, reference answers and available explanation.
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∫₀^(π) sin² x dx equals:Read the answer and worked solution →∫ dx/(a² - x²) (for |x| < a):Read the answer and worked solution →$\displaystyle\int \cos x\,dx$ equals:Read the answer and worked solution →To evaluate $\int x\cos x\, dx$, the best technique is:Read 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
∫ dx/(a² - x²) (for |x| < a):
- arcsin(x/a) + C
- (1/2a) ln |(a+x)/(a-x)| + C
- -arctan(x/a) + C
- ln |a-x| + C
Show answer and solution
Correct answer: B. (1/2a) ln |(a+x)/(a-x)| + C
**By partial fractions:** 1/(a² - x²) = 1/((a-x)(a+x)) = (1/2a)(1/(a-x) + 1/(a+x)).
**Integrating:** (1/2a) ln |(a+x)/(a-x)| + C. Note difference from ∫ dx/(a² + x²) = (1/a) arctan(x/a) + C.
$\displaystyle\int \cos x\,dx$ equals:
- $-\cos x + C$
- $\cos x + C$
- $\sin x + C$
- $-\sin x + C$
Show answer and solution
Correct answer: C. $\sin x + C$
Since d/dx(sin x) = cos x, ∫cos x dx = sin x + C.
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