Integral help please [Solved!]

X

can you help me solve this problem?

`int0^1 1/(1+x^4)dx`

Relevant page

<a href="/methods-integration/8-integration-trigonometric-substitution.php">8. Integration by Trigonometric Substitution</a>

What I've done so far

I've looked all over the techniques in the Methods of Integration chapter, but cant find anything that will do it.

X

Hi Garrett

It is possible, but it is pretty ugly.

You need to express `1+x^4` as

`(x^2+x sqrt2 + 1)(x^2-x sqrt2 +1)`

Then separate it into 2 fractions using partial fractions techniques:

<a href="/methods-integration/11-integration-partial-fractions.php">11. Integration by Partial Fractions</a>

Your final answer involves the log of the above expressions in brackets and arctan of a conjugate pair.

Personally, I would suggest approaching it using numerical integration:

<a href="/integration/6-simpsons-rule.php">6. Simpson&rsquo;s Rule</a>

It is less work, less agony and your final answer will have similar decimal place accuracy. But I guess you are required to use algebraic integration techniques, yes?

BTW, your final answer is around 0.87.

X

Thanks for your help sir, but I don't understand how you express `1+x^4` as `(x^2+x sqrt2 + 1)(x^2-x sqrt2 +1)`. Can you show me how to do that ?

(and yes, i have to do it algebraically)

X

Hi Garrett

To get an `x^4+1` term, it will involve the expansion of `(x^2 + 1)^2`. 

But when you do that, you don't quite get `(x^4+1)` because you will have `2x^2` that you don't want. You can remove that term by including the `+xsqrt2` and `-xsqrt2` terms as shown.

It might be easier to think of it as:

`[(x^2 + 1)+x sqrt2] [(x^2 +1) - x sqrt2]`

We are just using difference of 2 squares:

<a href="/factoring-fractions/2-common-factor-difference-squares.php">2. Common Factor and Difference of Squares</a>

Hope that helps.

X

ok, I got that part now.

For the integration, you said to do partial fractions.

`1/((x^2+x sqrt2 + 1)(x^2-x sqrt2 +1))`

`=A/(x^2+x sqrt2 + 1)+ B/ (x^2 - x sqrt2 + 1)`

So

`x^4+1 ` `= A(x^2 - x sqrt2 + 1) +B(x^2 + x sqrt2 + 1)`

But I'm stuck there.

X

There are a few problems. Once we do the cross multiply, we'll just have `1` on the LHS.

<b>Hint 1:</b> Our aim is to eventually get the number `1` on the RHS as well. (This is the `1` on the top of `1/(x^4+1)`. 

<b>Hint 2:</b> When you have the highest power of `x` in the denominator being 2, what will you put in the top of your 2 partial fractions?

Can you now set up an expression and then solve it for `A` and `B`?

X

Ah, I see. So I guess it's

`1/(x^4+1)=(Ax+B)/(x^2+x sqrt2 + 1)` `+ (Cx+D)/ (x^2 - x sqrt2 + 1)`


Multiplying out:

`1 ` `= (Ax+B)(x^2 -x sqrt2 + 1)` `+ (Cx+D) (x^2 + x sqrt2 + 1)`

`1 ` `= (Ax^3 -Ax^2 sqrt2 + Ax + Bx^2 -Bx sqrt2 + B)` `+ (Cx^3 + Cx^2 sqrt2 + Cx + Dx^2 + Dx sqrt2 + D)`

`1 ` `= (Ax^3 +(B-A sqrt2)x^2 + (A -B sqrt2)x + B)` `+ (Cx^3 + (C sqrt2 +D)x^2 + (C + D sqrt2)x + D)`

Then for the `x^3` terms:

`A + C =0` .... (1)

For the `x^2` terms:

`B - A sqrt2 + C sqrt2 + D = 0` .... (2)

For the `x` terms:

`A -B sqrt2 + C + D sqrt2 = 0` .... (3)

For the constant terms:

`B + D = 1` .... (4)

From (1), (2) and (4) I gott:

`A = 1/(2sqrt(2)`

`C = -1/(2sqrt(2)`

From (2), (3) and (4), I got

`B=1/2` and `D=1/2`

So 

`1/(x^4+1) ` `=(xsqrt(2) + 2)/(4(x^2+1+xsqrt(2)))` `+ (-xsqrt(2) + 2)/(4(x^2+1-xsqrt(2)))`

Now the integral:


`int 1/(x^4+1)dx ` `=int(xsqrt(2) + 2)/(4(x^2+1+xsqrt(2)))dx` `+ int(-xsqrt(2) + 2)/(4(x^2+1-xsqrt(2)))dx`

But I'm stuck again. What to do now?

X

It's very good what you've done.

In your final steps to tidy up our partial fractions, you did a bit too much work!

I expect you got this as one of your intermediate steps:

`1/(x^4+1) ` `=(x + sqrt(2))/(4sqrt(2)(x^2+1+xsqrt(2)))` `+ (-x + sqrt(2))/(4sqrt(2)(x^2+1-xsqrt(2)))`

Can you see it now?

X

Of course!

`int 1/(x^4+1)dx ` `=1/(4sqrt(2)) int(x + sqrt(2))/(x^2+1+xsqrt(2))dx` `+ 1/(4sqrt(2)) int(-x + sqrt(2))/((x^2+1-xsqrt(2)))dx`

`=1/(4sqrt(2))[ ln(x^2+1+xsqrt(2)):} ` `{:- ln(x^2+1-xsqrt(2))]`

`=1/(4sqrt(2))[ ln(x^2+1+xsqrt(2)) / ln(x^2+1-xsqrt(2))]`

For the definite integral we have:

`int0^1 1/(x^4+1)dx` `=1/(4sqrt(2))[ ln(x^2+1+xsqrt(2)) / ln(x^2+1-xsqrt(2))]0^1`

`=1/(4sqrt(2))[ ln(2 + sqrt(2)) / ln(2 - sqrt(2)) - 1]`

`=0.86697`

Thanks a lot for your help, sir!

X

You're welcome. Well done!