Mathematics

Evaluate:
$$\int { \sqrt { { x }^{ 2 }-16 }  }  \ dx$$


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Subjective Hard Published on 17th 09, 2020
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Q1 Single Correct Medium
The value $$\displaystyle \sqrt{2}\int\frac{\sin x dx}{\sin(x-\frac{\pi}{4})}$$ is
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  • B. $$\displaystyle x- \log|\cos(x-\frac{\pi}{4})|+C$$
  • C. $$\displaystyle x+ \log|\cos(x-\frac{\pi}{4})|+C$$
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Q2 Subjective Medium
Solve:
$$\int {\dfrac{{{x^3} - 4{x^2} + 6x + 5}}{{{x^2} - 2x + 3}}} dx$$

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Q3 Single Correct Hard
If $$ \displaystyle \int_{0}^{\infty }\frac{e^{-ax}\sin bx}{x}\:dx= \tan^{-1}\left ( b/a \right ) $$ then the value of $$ \displaystyle \int_{0}^{\infty }\frac{\sin ^2ax}{x^2}\:dx $$ equals, $$ a\neq 0 $$
  • A. $$ \displaystyle \frac{\pi a}{3} $$
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$$\int {\dfrac{{2 - 3\sin x}}{{{{\cos }^2}x}}dx} $$

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Q5 Passage Hard
Let us consider the integral of the following forms
$$f{(x_1,\sqrt{mx^2+nx+p})}^{\tfrac{1}{2}}$$
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Case II If $$p>0$$, then put $$\sqrt{mx^2+nx+C}=u\pm \sqrt{p}$$
Case III If quadratic equation $$mx^2+nx+p=0$$ has real roots $$\alpha$$ and $$\beta$$, then put $$\sqrt{mx^2+nx+p}=(x-\alpha)u\:or\:(x-\beta)u$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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