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NDA 2015
Exam
NDA
Year
2015
Subject
Mathematics
Chapter
Definite Integration

NDA 2015 · Mathematics · Definite Integration

The value of ∫_a^b (x^7+ x)/( x) d x where a+b=0 is

The value of ∫abx7+sin⁡xcos⁡xdx\int_a^b \frac{x^7+\sin x}{\cos x} d x where a+b=0a+b=0 is

Options

  1. A2b−asin⁡(b−a)2 \mathrm{ b}-\mathrm{a} \sin (\mathrm{b}-\mathrm{a})
  2. Ba+3bcos⁡(b−a)a+3 b \cos (b-a)
  3. Csin⁡a−(b−a)cos⁡b\sin \mathrm{a}-(\mathrm{b}-\mathrm{a}) \cos \mathrm{b}
  4. D0

Correct answer

D. 0

Step-by-step solution

a+b=0⇒a=−bI=∫−bbx7+sin⁡xcos⁡x\begin{aligned} & a+b=0 \Rightarrow a=-b \\ & I=\int_{-b}^b \frac{x^7+\sin x}{\cos x} \end{aligned} Using property ∫−aaf(x)dx={∫02af(x)dx; if f(x) is even 0; if f(x) is odd f(x)=x7+sin⁡xcos⁡xf(−x)=(−x)7+sin⁡(−x)cos⁡(−x)=−x7−sin⁡xcos⁡x=−[x7+sin⁡xcos⁡x]\begin{aligned} & \int_{-a}^a f(x) d x= \begin{cases}\int_0^{2 a} f(x) d x & ; \text{ if } f(x) \text{ is even } \\ 0 & ; \text{ if } f(x) \text{ is odd }\end{cases} \\ & f(x)=\frac{x^7+\sin x}{\cos x} \\ & f(-x)=\frac{(-x)^7+\sin (-x)}{\cos (-x)}=\frac{-x^7-\sin x}{\cos x} \\ & =-\left[\frac{x^7+\sin x}{\cos x}\right] \end{aligned}=-\mathrm{f}(\mathrm{x})SoSof(x)isoddhenceis odd hence\mathrm{I}=0$
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