Mathematics: Calculus 2 master needed; evaluate the integral PLEASE SHOW STEPS IF IM WRONG I split off the sin^3 so i can use the pythag identity and allows for u substitution u=cosx du=-sinx dx -du=sin dx I move the negative towards the outside of the integral. then i divide the terms by sqroot 2||| I eventually get to a=1/2 b =5/2 did I miss anything? Or is this the final answer?

Calculus 2 master needed; evaluate the integral

Ответ:

devaughnnorthcu8221

01.02.2022

Your process is indeed correct!

The full solution is:

$\displaystyle \int\frac{\sin ^3 x}{\sqrt{\cos x}}\, dx= \frac{2}{5} \cos^{{}^{5}\!/\!{}_{2}} x - 2\cos ^{{}^{1}\!/\!{}_{2}}x + C$

Step-by-step explanation:

We want to evaluate the integral:

$\displaystyle \int \frac{\sin^3(x)}{\sqrt{\cos(x)}}\, dx$

As you had done, we can rewrite our integral as:

$\displaystyle =\int \frac{\sin(x)(\sin^2(x))}{\sqrt{\cos(x)}}\, dx$

Using the Pythagorean Identity, this is:

$\displaystyle =\int \frac{\sin(x)(1-\cos^2(x))}{\sqrt{\cos(x)}}\, dx$

Now, we can make a substitution. Let u = cos(x). Then:

$\displaystyle du = - \sin x \, dx$

Substitute:

$\displaystyle = \int\frac{1-u^2}{\sqrt{u}} \, \left(- du\right)$

Simplify:

$\displaystyle = -\int \frac{1}{\sqrt{u}} - \frac{u^2}{\sqrt{u}}\, du$

Rewrite:

$\displaystyle = -\int u^{{}^{-1}\!/\!{}_{2}} - u^{{}^{3}\!/\!{}_{2}}\, du$

By the Reverse Power Rule:

$\displaystyle = -\left(2u^{{}^{1}\!/\!{}_{2}} - \frac{2}{5} u^{{}^{5}\!/\!{}_{2}}\right) + C$

Simplify:

$\displaystyle = \frac{2}{5} u^{{}^{5}\!/\!{}_{2}}-2u^{{}^{1}\!/\!{}_{2}} + C$

Back-substitute:

$\displaystyle = \frac{2}{5} \cos^{{}^{5}\!/\!{}_{2}} x - 2\cos ^{{}^{1}\!/\!{}_{2}}x + C$

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Ответ:

bluenblonderw

01.02.2022

Mathematics

= - 2 $\sqrt{cos(x)}$ + 2 cos⁵/₂ (x) + C

Step-by-step explanation:

∫ sin³ (x) dx

$\sqrt{cos(x)}$

= ∫ sin² (x) sin (x) dx

$\sqrt{cos(x)}$

= ∫ (1 - cos² (x) sin (x) dx

$\sqrt{cos(x)}$

= ∫ - 1 - u² du

√u

= ∫ - 1 + u³/₂ du

√u

= - ∫ 1 du + ∫ u³/₂ du

√u

substitute it back

= - 2 √u + 2 cos⁵/₂ (x)

add constant, therefore

= - 2 $\sqrt{cos(x)}$ + 2 cos⁵/₂ (x) + C

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Ответ:

smilxess

15.03.2020

Mathematics

I think that it might be C. but I am NOT entirely sure though.

Step-by-step explanation:

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