Prove the following Exercise: ^ 1 _ -1 x^ 17 ^ 4 x dx=0 - Vidyadip

Question

Prove the following Exercise:
$\int^{1}_{-1}\text{x}^{17}\cos^{4}\text{x}\ \text{dx}=0$

✓

Answer

$\text{Let I}=\int^{1}\limits_{-1}\text{x}^{17}\cos^{4}\text{x dx}$
Also, let $\text{f(x)}=\text{x}^{17}\cos^{4}\text{x}$
$\Rightarrow\text{f}\ (-\text{x)}=(-\text{x)}^{17}\cos^{4}(-\text{x)}=-\text{x}^{17}\cos^{4}\text{x}=-\text{f (x)}$
Therefore, f(x) is an odd function.
it is know that if f(x) is an odd function, then $\int^{\text{a}}\limits_{-\text{a}}\text{f (x) dx}=0$
$\therefore\text{I}=\int^{1}\limits_{-1}\text{x}^{17}\cos^{4}\text{x dx}=0$
Hence, the given result is Proved.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "2 Marks Questions" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Construct a $2 \times 3$ matrix $A = [a_{ij}]$ whose elements $a_{ij}$ are give by:
$a_{ij} = i.j$

If $A$ is a square matrix of order $3$ with determinant $4$, then write the value of $|-A|.$

Find the unit vector in the direction of vector $\overrightarrow {PQ} $ , where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer:
$f: R → R$ defined by $f(x) = 3 – 4x$

Find all the points of discontinuity of the greatest integer function defined by f(x) = [x], where [x] denotes the greatest integer less than or equal to x.

If $\vec{\text{a}}.\vec{\text{a}}=0$ and $\vec{\text{a}}.\vec{\text{b}}=0,$ what can you conclude about the vector $\vec{\text{b}}$?

Find $\lambda,$ if $\big(2\hat{\text{i}}+6\hat{\text{j}}+14\hat{\text{k}}\big)\times\big(\hat{\text{i}}-\lambda\hat{\text{j}}+7\hat{\text{k}}\big)=\vec{0}.$

Find the slope of the tangent to curve $y = x^3 – x + 1$ at the point whose x-coordinate is $2$.

If $\text{y}=\cot\text{x}$ show that $\frac{\text{d}^2\text{y}}{\text{dx}^2}+2\text{y}\frac{\text{dy}}{\text{dx}}=0$

In answering a question on a multiple choice test, a student either knows the answer or guesses. Let $\frac{3}{4}$ be the probability that he knows the answer and $\frac {1}{4}$ be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability $\frac{1}{4}$. What is the probability that the student knows the answer given that he answered it correctly?