Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

For example, I recently came across the following way to evaluate the integral of $\cos^2 x - \sin^2 x$ without using double angle formulas:

$$\int dx (\cos^2 x - \sin^2 x) = \int dx(\cos x + \sin x)(\cos x - \sin x) = \int u du = \frac{u^2}{2} + C$$

Where $u = \cos x + \sin x$. One can expand the final result to get $\sin x \cos x + C'$ as the final result, i.e. $\frac{1}{2} \sin (2x) + C'$. Though this may take longer, I find this solution valuable because it reminds us that there is more than one solution, even to a seemingly rigid problem like this.

Another example is solving $\lim \limits_{n \to \infty} \sqrt[n]{n}$ using AM-GM and Squeeze theorem:

$\frac{n - 2 + 2 \sqrt{n}}{n} \geq \sqrt[n]{n} \geq 1$ by definition

$1 - \frac{2}{n} + \frac{2}{\sqrt{n}} \geq \sqrt[n]{n} \geq 1$ and then squeeze.

I found this solution to be much more interesting than the standard solution which is to take $\ln$ of the expression and find that limit.

Can you show me other examples of this? (I am personally only an advanced high schooler, but I will certainly appreciate answers at any level and hopefully I will be able to fully understand them some time in the future.) Also note that any area of study is acceptable.

share|improve this question

1 Answer 1

Perhaps the most standard way to compute the sum $$1+3+5+\ldots +(2n-1)=n^2$$ would be to use the arithmetic series with $a_0=1$, $d=2$.

Here is another solution:

enter image description here

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.