Take the integral: ∫3x2a2−x2dx Factor out constants: ∫3x2a2−x2dx For the integrand x2a2−x2, (assuming all variables are positive) substitute x=asin⁡(u)anddx=acos⁡(u)du. Then a2−x2=a2−a2sin2(u)=acos⁡(u)andu=sin−1(xa): =3a∫a3sin2(u)cos2(u)du Factor out constants: =3a4∫sin2(u)cos2(u)du Write cos2(u)as1−sin2(u): =3a4∫sin2(u)(1−sin2(u))du Expanding the integrand sin2(u)(1−sin2(u)) gives sin2(u)−sin4(u): =3a4∫(sin2(u)−sin4(u))du Integrate the sum term by term and factor out constants: =−3a4∫sin4(u)du+3a4∫sin2(u)du Use the reduction formula, ∫sinm(u)du=−cos⁡(u)sinm−1(u)m+m−1m∫sin−2+m(u)du, where m = 4: =34a4sin3(u)cos⁡(u)+3a44∫sin2(u)du Write sin2(u)as12−12cos⁡(2u):

"∫0a3x2a2−x2dx" was answered by students like you

mattgondek4

Answered question

2021-09-02

$\int_{0}^{a} 3 x^{2} \sqrt{a^{2} - x^{2}} d x$

Answer & Explanation

Bertha Stark

Skilled2021-09-03Added 96 answers

Take the integral:
$\int 3 x^{2} \sqrt{a^{2} - x^{2}} d x$
Factor out constants:
$\int 3 x^{2} \sqrt{a^{2} - x^{2}} d x$
For the integrand $x^{2} \sqrt{a^{2} - x^{2}}$ , (assuming all variables are positive) substitute $x = a \sin (u) and d x = a \cos (u) d u$ . Then $\sqrt{a^{2} - x^{2}} = \sqrt{a^{2} - a^{2} \sin^{2} (u)} = a \cos (u) and u = \sin^{- 1} (\frac{x}{a})$ :
$= 3 a \int a^{3} \sin^{2} (u) \cos^{2} (u) d u$
Factor out constants:
$= 3 a^{4} \int \sin^{2} (u) \cos^{2} (u) d u$
Write $\cos^{2} (u) a s 1 - \sin^{2} (u)$ :
$= 3 a^{4} \int \sin^{2} (u) (1 - \sin^{2} (u)) d u$
Expanding the integrand $\sin^{2} (u) (1 - \sin^{2} (u))$ gives $\sin^{2} (u) - \sin^{4} (u)$ :
$= 3 a^{4} \int (\sin^{2} (u) - \sin^{4} (u)) d u$
Integrate the sum term by term and factor out constants:
$= - 3 a^{4} \int \sin^{4} (u) d u + 3 a^{4} \int \sin^{2} (u) d u$
Use the reduction formula, $\int \sin^{m} (u) d u = - \frac{\cos (u) \sin^{m - 1} (u)}{m} + \frac{m - 1}{m} \int \sin^{- 2 + m} (u) d u$ , where m = 4:
$= \frac{3}{4} a^{4} \sin^{3} (u) \cos (u) + \frac{3 a^{4}}{4} \int \sin^{2} (u) d u$
Write $\sin^{2} (u) a s \frac{1}{2} - \frac{1}{2} \cos (2 u)$ :

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