Is there an easier technique to integrate this

[turutk]

Homework Statement

$$\int\sqrt{sinx}*cos^3x$$

Homework Equations

The Attempt at a Solution

by writing as root(sinx) (1-sin^2(x)) (cosx) i can solve but since the worksheet is called basic techniques of integration i am looking for an easier and more obvious solution if possible.

also can this be integrated by the way i wrote at 3
$$\int\sqrt{tanx}*csc^2x$$

 

[Dick]

Substituting u=sin(x) into that is pretty easy and basic, isn't it?

 

[micromass]

Apply substitution u=sin(x).

 

[zhermes]

using 1-sin^2 is the only way I can see of doing it... why's that not easy?

 

[turutk]

since i am very new to the topic i have no idea how to do integration by substitution. we are trying to work derivatives backwards.

 

[Mark44]

Homework Statement

$$\int\sqrt{sinx}*cos^3x$$

Homework Equations

The Attempt at a Solution

by writing as root(sinx) (1-sin^2(x)) (cosx) i can solve but since the worksheet is called basic techniques of integration i am looking for an easier and more obvious solution if possible.

The way you found is probably the easiest and most obvious approach, using an ordinary substitution u = sin(x), du = cos(x)dx[/color]. BTW, you omitted dx. In the easy integrals you start with, omitting this won't cause problems, but if you continue to omit the differential in other techniques, it will definitely cause problems for you.

After making the substitution, the integral becomes
$$\int u^{1/2}(1 - u^2)du = \int u^{1/2} du - \int u^{5/2} du$$

also can this be integrated by the way i wrote at 3
$$\int\sqrt{tanx}*csc^2x$$

Sort of, but I would do a little work on the integrand before tackling the integration. I would rewrite the integral as
$$\int \frac{1}{\sqrt{cot(x)}}*csc^2(x)\bold{dx}$$

My substitution would be u = cot(x), du = -csc²(x)dx

 

[turutk]

okay i got it. thank you all for your help