How do I integrate the function $\sqrt{(6x + 2)}$?

How do you integrate $$\sqrt{(6x + 2)}$$?

I've tried to use the following substitutions: let $$x = \sin(u)$$ and $$dx = \cos(u)$$ (along the lines of the Yahoo Answers link). I tried looking for simple examples of integrals with square roots on Yahoo Answers and elsewhere by Googling, but couldn't find any simpler ones, and that substitution got me nowhere.

• Welcome to Math.Stackexchange! What approaches have to tried so far? – user458276 Jan 5 at 2:07
• Do you know how to integrate $\sqrt{x}$? Have you heard of "u-substitution"? – JMoravitz Jan 5 at 2:08
• I don't know if my search engine is bad, but I just found out complicated examples. This is a simple one – bcloney Jan 5 at 2:12
• Integrating $\sqrt x$ is basically the same as integrating any power: $$\int x^n dx = \frac{1}{n+1}x^{n+1} + C$$ Keep in mind $\sqrt x = x^{1/2}$. – Eevee Trainer Jan 5 at 2:17
• The technique of "u-substitution" for integration is just the same thing as the technique of using the chain-rule for derivation in reverse. Similarly, finding antiderivatives are essentially analogous to finding derivatives in reverse. Even if it is your first week with integration, I would expect you to have some experience with derivatives and should have seen how to derive $x^{3/2}$. If you know how to derive $x^{3/2}$ then you should be able to figure out how to integrate $x^{1/2}$. – JMoravitz Jan 5 at 2:25

Hints:

• Make the $$u$$-substitution $$u = 6x+2$$.
• Don't forget that $$\sqrt u = u^{1/2}$$ and that, for all $$n \neq -1$$, we have

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$

You are looking for:

$$\int(6x+2)^\frac12 dx$$

Notice that if we let $$u=6x+2$$, $$\frac{du}{dx}=6$$ which leads to $$dx=\frac16 du$$

In other words, the above integral is exactly the same as this: $$\int (u)^\frac12 \cdot \frac 16 du$$ You can take the constant outside the integral to make this: $$\frac 16 \int u^\frac 12 du$$

And deal with that integral using the normal power rules.

At the end, don't forget to resubsitute $$u=6x+2$$ back in!

$$\int\sqrt{6x+2}\operatorname dx=\frac16\cdot\frac23(6x+2)^{\frac32}+C=\frac19(6x+2)^{\frac32}+C$$, by using the power rule for derivatives and the chain rule.