Let g(t) = $\int_{- π / 2}^{π / 2} c o s (\frac{π}{4} t + f (x)) d x,$ where f(x) = log_e $(x + \sqrt[]{x^{2} + 1}), x \in R .$ Then which one of the following is correct?

Option 1 -

g(1) = g(0)

Option 2 -

g(1) = $\sqrt[]{2} g (0)$

Option 3 -

$g (1) + g (0) = 0$

Option 4 -

$\sqrt[]{2} g (1) = g (0)$

3 Views | Posted a month ago

Asked by Shiksha User

1 Answer

Answered by

Vishal Baghel | Contributor-Level 10

a month ago

Correct Option - 4

Detailed Solution:

$? f (x) = l n (x + \sqrt[]{1 + x^{2}}) ∴ f (- x) = - f (x)$ .

Hence f(x) is an odd function.

$N o w g (t) = \int_{- \frac{π}{2}}^{\frac{π}{2}} c o s (\frac{π}{4} t + f (x)) d x$

Put t = 0, g(0) = $\int_{- \frac{π}{2}}^{\frac{π}{2}} c o s (f (x)) d x = 2 \int_{0}^{\frac{π}{2}} c o s (f (x)) d x . . . . . . . . . . . . . . . (i)$

where cos(f(x)) is an even function.

Now again put t = 1,

$∴ g (1) = \frac{1}{\sqrt[]{2}} \times g (0), f r o m (i)$

$∴ g (0) = \sqrt[]{2} g (1)$

<math> <mrow> <mo>?</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mroot> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow></mrow> </mroot> </mrow> <mo>)</mo> </mrow> <mo>∴</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> .Hence f(x) is an odd function. <math> <mrow> <mi>N</mi> <mi>o</mi> <mi>w</mi> <mtext> </mtext> <mtext> </mtext> <mi>g</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mstyle displaystyle="true"> <mrow> <munderover> <mo>∫</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mrow> <mfrac> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </munderover> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>π</mi> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <mi>t</mi> <mo>+</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>x</mi> </mrow> </mrow> </mstyle> </mrow> </math> Put t = 0, g(0) = <math> <mrow> <mstyle displaystyle="true"> <mrow> <munderover> <mo>∫</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mrow> <mfrac> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </munderover> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>2</mn> </mrow> </mrow> </mstyle> <mstyle displaystyle="true"> <mrow> <munderover> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mrow> <mfrac> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </munderover> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>x</mi> </mrow> </mrow> </mstyle> <mtext> </mtext> <mtext> </mtext> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> where cos(f(x)) is an even function.Now again put t = 1, <math> <mrow> <mo>∴</mo> <mi>g</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mroot> <mrow> <mn>2</mn> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> <mo>×</mo> <mi>g</mi> <mrow> <mo>(</mo> <mrow> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mi>f</mi> <mi>r</mi> <mi>o</mi> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math>  <math> <mrow> <mo>∴</mo> <mi>g</mi> <mrow> <mo>(</mo> <mrow> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mroot> <mrow> <mn>2</mn> </mrow> <mrow></mrow> </mroot> <mi>g</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </math>

0 Upvotes 0 Downvotes

Let g(t) = $\int_{- π / 2}^{π / 2} c o s (\frac{π}{4} t + f (x)) d x,$ where f(x) = log_e $(x + \sqrt[]{x^{2} + 1}), x \in R .$ Then which one of the following is correct?

1 Answer

Similar Questions for you

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question

Let g(t) = ∫ − π / 2 π / 2 c o s ( π 4 t + f ( x ) ) d x , where f(x) = loge ( x + x 2 + 1 ) , x ∈ R . Then which one of the following is correct?

1 Answer

Similar Questions for you

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question

Let g(t) = $\int_{- π / 2}^{π / 2} c o s (\frac{π}{4} t + f (x)) d x,$ where f(x) = log_e $(x + \sqrt[]{x^{2} + 1}), x \in R .$ Then which one of the following is correct?