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.

Link to view Solution of my question in image format:

enter image description here

Solution is correct. Kindly Explain the First Step, encircled in red color.

share|improve this question
2  
You evaluate the integral first. Assuming $p\ne1$, an antiderivative of ${1\over x^p}=x^{-p}$ is ${1\over -p+1}x^{-p+1}={1\over1-p}{1\over x^{p-1}}$. –  David Mitra Feb 5 '13 at 13:59
add comment

1 Answer 1

The first step is saying that an antiderivate of the function $\displaystyle\frac{1}{x^p}$ is $\displaystyle\frac{1}{1-p}\frac{1}{x^{p-1}}$ which you can check by verifying $$\frac{d}{dx}\left(\frac{1}{1-p}\frac{1}{x^{p-1}}\right) = \frac{1}{x^p}.$$

If $F$ is an antiderivative of $f$ (that is $F' = f$), then (one half of) the Fundamental Theorem of Calculus says that $\int_a^bf(x)dx = [F(x)]_a^b = F(b) - F(a)$.

Added Later: As David Mitra points out, the first paragraph only applies when $p \neq 1$.

share|improve this answer
add comment

Your Answer

 
discard

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.