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.

I am trying to figure how to solve the following antiderivative.

$$ \int (5x+3)^7 dx \\ $$ I've seen the step-by-step solution by WolframAlpha however what they are doing in this part:

For the integrand $(5x+3)^7$, substitute $$ u = 5x + 3 \\ du = 5 dx \\ $$

Why are they derivating $u$ ?

Thanks.

share|improve this question
2  
Are you familiar with integration by substitution? en.wikipedia.org/wiki/Integration_by_substitution –  anonymous Dec 2 '12 at 16:55
    
Yes, I am, I understand now, I had no simple example –  Francis Dec 2 '12 at 17:02

2 Answers 2

up vote 2 down vote accepted

The general theorem is like this:

$$\int (f\circ g)(x)g'(x)dx=\int f(u)du \Leftrightarrow u=g(x).$$

It may help to note that in calculations, we have $u=\text{something}$ and $du=(\text{something})'dx$. This is because of the definition of the differential. That is, $dy=f'(x)dx$ if and only if $y=f(x)$. (The definition of the differential just comes from the "multiplying" of $dx$ to both sides in $\frac{dy}{dx}=f'(x)$.)

share|improve this answer

Directly:

$$\int (5x+3)^7dx=\frac{1}{5}\int 5(5x+3)^7dx=\frac{1}{5}\int(5x+3)^7d(5x+3)=$$

$$=\frac{1}{5}\frac{(5x+3)^8}{8}+C=\frac{(5x+3)^8}{40}+C$$

share|improve this answer

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.