Integration help?

Question

How do I integrate:

3
-------
sqrt(t +1)

I'm terrible at u-sub. I took the 3 out and now I need to integrate 1 / sqrt(u). I'm not sure how to do that. (or if that's even right)

Answer 1

Given 3/√(t + 1), find its integral.

Let u = t + 1.

Then √(t + 1) = (t + 1)^1/2 = u^1/2, 3/(t + 1)^1/2 = 3/u^1/2 = 3 u^(-1/2) and du = dt.

Now the integrand is in the integrable form:

ƒ k u^(-1/2) du = k ƒ u^(-1/2) du, where k = 3.

k ƒ u^(-1/2) du = k [u^(-1/2 + 2/2)]/(1/2) + c = 2k u^1/2 + c.

Remember, when we divide by a fraction, we multiply the dividend (the number being divided) by its reciprocal. For example, a/(1/2) = (2/1)a = 2a. That's how we got the final form of the integral.

Now all that remains is to back substitute all of our values and get the final answer.

k = 3 and u = (t + 1).

So k ƒ u^(-1/2) du = 2k u^1/2 + c = 2(3) (t + 1)^1/2 + c =
6 (t + 1)^1/2 + c

Answer 2

You don't need to make a substitution because the contents of the bracket are linear so will differentiate to a number. You can therefore integrate using the power rule if you express sqrt as power 0.5 or 1/sqrt as power -0.5. Differentiate what you think is the answer to be sure that you have got the multiplying number correct.

Answer 3

the integral of 3/sqrt(t+1) = 3 [ 2(sqrt 1 + t) ] = 6 sqrt (1 + t)

the formula is 1/sqrt (a + bx) = 2 sqrt ( a + bx)/b

Hope this helps. If they still sell copies of the CRC Math Tables by Dr. Sam Selby, I'd recommend you get a copy. Selby was my professor.

Answer 4

the three can go outside the integral

and we can say 1/sqrt(t+1) = (t+1)^-0.5 which is easier to integrate and it becomes 2(t+1)^0.5 +k (k the constant of integration).

Recombining with the 3 it is 3(2(t+1)^0.5 + k) = 6(t+1)^0.5 + C where C = 3k

Answer 5

Worst of all, you don't need to substitute anything. This is
3*(t+1)^(-1/2) which (by the exponent rule) becomes
3* 2* (t+1)^(1/2) + C = 6√(t+1) + C

HTH ☺


Doug

Answer 6

1/sqrt(u) = u^(-.5)
integrate u^(-.5)
that equals 2u^(.5).

so the answer would be 6(t+1)^(.5)


I forgot my +C, oops.
my teacher would kill me for that!