Part 2: Evaluate the integrals. 3. ∫(e^x)/((1+e^x)^2) dx; 4....

Transcript

00:01 You want to evaluate this integral.

00:02 So i noticed that in the denominator, i got e to the 4x, which is the square of e to the 2x.

00:10 So it suggests making a substitution that looks like e to the 2x.

00:18 And because of the numbers in the denominator, i'm going to choose it like this.

00:30 Excuse me.

00:32 The other way around.

01:02 So we got to have a 5 4s.

01:13 And then the numerator, we just have du.

01:19 And then the denominator, we got, okay, i hope that's a square root.

01:39 Bring the 25 outside the square root, we get 1 fourth this integral, okay? and everybody knows that that integral is an inverse sine function.

02:03 So there's that one.

02:04 So i'm going to this integral here's what i'm going to do i'm going to write x to the fourth like this some constant a plus another constant b so let's expand that out all right and so we want this to just equal x to the fourth all right so we have to get the square and so forth fourth terms to cancel out.

03:11 That says a is minus 6, and then we got 9 plus 3a plus b equals 0.

03:25 So b is 9, okay? so this is this integral, and this is squared, all right? and so this becomes, comes.

04:30 All right.

04:32 So we get one third x cubed.

04:36 This is minus three x.

04:40 And then we get nine.

04:45 So i'm going to mess around with this denominator.

04:48 Basically inside that square root, i'm going to add and subtract four.

04:55 So i'm going to get something that looks like this...

Question

Part 2: Evaluate the integrals. 3. $\int \frac{e^x}{(1+e^x)^2} dx$; 4. $\int \frac{1}{x+x\sqrt{x}} dx$; 5. $\int x\sqrt{1-x^2} dx$; 6. $\int \frac{\sin x}{1-\cos x} dx$.

Question

Part 2: Evaluate the integrals. 3. $\int \frac{e^x}{(1+e^x)^2} dx$; 4. $\int \frac{1}{x+x\sqrt{x}} dx$; 5. $\int x\sqrt{1-x^2} dx$; 6. $\int \frac{\sin x}{1-\cos x} dx$.

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