Indian Navy SSR Definite Integrals — practice questions

Explore 2 free Definite Integrals questions in Indian Navy SSR Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Definite Integrals questions and exercises with answers

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If $f$ is **odd** on $[-a, a]$, then $\displaystyle\int_{-a}^{a} f(x)\, dx$ equals:Read the answer and worked solution →$\displaystyle\int_0^{\infty} \dfrac{\ln x}{1 + x^2}\, dx = ?$Read the answer and worked solution →

Worked practice questions

If $f$ is odd on $[-a, a]$, then $\displaystyle\int_{-a}^{a} f(x)\, dx$ equals:

$0$
$2\int_0^a f(x)dx$
$a$
Cannot be determined

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Correct answer: A. $0$

By symmetry, odd-function integrals over symmetric intervals vanish. (For even $f$: $2\int_0^a f$.)

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$\displaystyle\int_0^{\infty} \dfrac{\ln x}{1 + x^2}\, dx = ?$

Diverges
$0$
$\pi/4 \cdot \ln 2$
$\ln(\pi/2)$

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Correct answer: B. $0$

Split at $x = 1$ and substitute $x = 1/u$ in the part from $1$ to $\infty$: $\int_1^\infty \ln x/(1+x^2)\,dx = \int_0^1 -\ln u/(1+u^2)\,du$. **So total = $\int_0^1 \ln x/(1+x^2)dx + \int_0^1 -\ln u/(1+u^2)du = 0$.**