1. ## Check the answer of integration(Definite)

The value of $\displaystyle \int_0^{\infty}\{\frac{dx}{(a^2+x^2)^7}\}$ is equal to ?

My try:

Put $\displaystyle x=a tan(\theta)$

$\displaystyle dx =asec^2(\theta)d\theta$

Thus integration becomes

$\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{a sec^2(\theta)d\theta}{a^{14}~(1+tan^2(\theta))^7}$

Using $\displaystyle 1+tan^2(h) = sec^2(h)$

$\displaystyle \frac{1}{a^{13}}\int_{0}^{\frac{\pi}{2}}\{\frac{d\ theta}{(sec^{12}(\theta)}\}$

$\displaystyle \frac{1}{a^{13}}\int_{0}^{\frac{\pi}{2}}\{cos^{12} (\theta)d\theta\}$

Now here I have used the formula that for even "n"

$\displaystyle \int_{0}^{\pi /2} (cos^{n}(xdx)) = \frac{n-1}{n} \cdot \frac{n-3}{n-2} ....\frac{1}{2}\cdot \frac{\pi}{2}$

$\displaystyle \frac{1}{a^{13}}(\frac{11}{12}\cdot \frac{9}{10}...\frac{1}{2}\cdot \frac{\pi}{2})$

My answer thus was $\displaystyle \frac{363}{2048a^{13}}$

Options are

• $\displaystyle \frac{231}{2048a^{13}}$

• $\displaystyle \frac{235}{2048a^{13}}$

--------------------------
Where did I go wrong.

2. ## solution

method is good -- 11/12*9/10*7/8*5/6*3/4*1/2*π/2 = *pi

The value of $\displaystyle \int_0^{\infty}\{\frac{dx}{(a^2+x^2)^7}\}$ is equal to ?

My try:

Put $\displaystyle x=a tan(\theta)$

$\displaystyle dx =asec^2(\theta)d\theta$

Thus integration becomes

$\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{a sec^2(\theta)d\theta}{a^{14}~(1+tan^2(\theta))^7}$

Using $\displaystyle 1+tan^2(h) = sec^2(h)$

$\displaystyle \frac{1}{a^{13}}\int_{0}^{\frac{\pi}{2}}\{\frac{d\ theta}{(sec^{12}(\theta)}\}$

$\displaystyle \frac{1}{a^{13}}\int_{0}^{\frac{\pi}{2}}\{cos^{12} (\theta)d\theta\}$

Now here I have used the formula that for even "n"

$\displaystyle \int_{0}^{\pi /2} (cos^{n}(xdx)) = \frac{n-1}{n} \cdot \frac{n-3}{n-2} ....\frac{1}{2}\cdot \frac{\pi}{2}$

$\displaystyle \frac{1}{a^{13}}(\frac{11}{12}\cdot \frac{9}{10}...\frac{1}{2}\cdot \frac{\pi}{2})$

My answer thus was $\displaystyle \frac{363}{2048a^{13}}{\color{red}\,\pi}$

Options are

• $\displaystyle \frac{231}{2048a^{13}}$

• $\displaystyle \frac{235}{2048a^{13}}$

--------------------------
Where did I go wrong.
First, You missed a $\displaystyle \pi$ in your solution.

Your mistake is in the line before you told us your answer. When you multiply it out, you should have $\displaystyle \frac{1}{a^{13}}\frac{11\cdot9\cdot7\cdot5\cdot3\c dot1\cdot\pi}{12\cdot10\cdot8\cdot6\cdot4\cdot2\cd ot2}=\boxed{\frac{231\pi}{a^{13}2048}}$

4. Originally Posted by Calculus26

method is good -- 11/12*9/10*7/8*5/6*3/4*1/2*π/2 = *pi
My calculated answer was wrong its supposed to be

$\displaystyle \frac{363}{1024a^{13}}$

Taking pi as 22/7, The answer will be correct now . thanks

Chris:No I didn't miss the pi, it was to be converted in accordance to the options. Though I messed up the calculation part.

So both the options are incorrect as "pi" is missing in them. Thanks to both of you for verifying

### how to check answer of definite integration

Click on a term to search for related topics.