I am trying to integrate (3x)(1+(x^2))^(1/2). Using integration by parts, I get u as 3x, du as 3 dx and dv comes as (1+(x^2)). To find v, I integrate dv and v comes out to be (1+(x^2))^(3/2) divided by 3x. I use the formula for integration by parts and get the answer. But it is not the correct answer. Why? http://www.wolframalpha.com/input/?i=Integrate+%283x%29%28%281%2Bx%5E%282%29%29%5E%281%2F2%29

Question

kropot72 · Answer

try as a solution (1 + x^2)^(3/2)
When this is differentiated by the chain rule the original expression is obtained.Therefore $$(1 + x ^{2})^{3/2}$$is the answer.

kropot72 · Answer

The above method is integration by inspection using the standard formula for integration with respect to (1 + x^2) and initially ignoring the term 3x.

anonymous · Answer

I didn't understand that. :(

kropot72 · Answer

Do you follow the explanation?

anonymous · Answer

I haven't heard of this method before but it sure seems interesting. Why are completely leaving out 3x?

kropot72 · Answer

Because it will reappear when the trial result is differentiated to check that the result is correct. Note: Integration by inspection is not appropriate in many cases. However in this case it gives a quick result.

campbell_st · Answer

if 

$$3 \int\limits x \sqrt{1 + x^2} dx$$

use integration by substitution for this


let $$u = x^2 + 1$$

$$\frac{du}{dx} = 2x$$

du = 2x dx

$$\frac{3}{2}\int\limits u^{1/2} du$$

$$\frac{3}{2}\int\limits\limits u^{1/2} du =\frac{3}{2} \times \frac{2}{3} u^{3/2} + C$$

giving $$F(x) = (x+1)^{3/2} + C$$