What is the integral of (a^2-x^2)^{5/2} ?

The integral of (a^2-x^2)^{5/2} is

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integral of (a^2-x^2)^{5/2}

\int (a^{2} - x^{2})^{\frac{5}{2}} d x

\frac{1}{6} x (a^{2} - x^{2})^{\frac{5}{2}} + \frac{5 a ^{2} x ( - x ^{2} + a ^{2} ) ^{\frac{3}{2}}}{24} + \frac{5 a ^{6}}{32} (2 arcsin (\frac{x}{a}) + sin (2 arcsin (\frac{x}{a}))) + C

Solution steps

\int (a^{2} - x^{2})^{\frac{5}{2}} d x

Apply Trigonometric Substitution

: \int a^{6} cos^{6} (u) d u

= \int a^{6} cos^{6} (u) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= a^{6} \cdot \int cos^{6} (u) d u

Apply Integral Reduction

= a^{6} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} \cdot \int cos^{4} (u) d u)

\int cos^{4} (u) d u = \frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u))

= a^{6} (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u))))

Substitute back

u = arcsin (\frac{1}{a} x)

= a^{6} (\frac{sin ( arcsin ( \frac{1}{a} x ) ) cos ^{5} ( arcsin ( \frac{1}{a} x ) )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (arcsin (\frac{1}{a} x)) sin (arcsin (\frac{1}{a} x)) + \frac{3}{8} (arcsin (\frac{1}{a} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{a} x)))))

Simplify

a^{6} (\frac{sin ( arcsin ( \frac{1}{a} x ) ) cos ^{5} ( arcsin ( \frac{1}{a} x ) )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (arcsin (\frac{1}{a} x)) sin (arcsin (\frac{1}{a} x)) + \frac{3}{8} (arcsin (\frac{1}{a} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{a} x))))) : \frac{1}{6} x (a^{2} - x^{2})^{\frac{5}{2}} + \frac{5 a ^{2} x ( - x ^{2} + a ^{2} ) ^{\frac{3}{2}}}{24} + \frac{5 a ^{6}}{32} (2 arcsin (\frac{x}{a}) + sin (2 arcsin (\frac{x}{a})))

= \frac{1}{6} x (a^{2} - x^{2})^{\frac{5}{2}} + \frac{5 a ^{2} x ( - x ^{2} + a ^{2} ) ^{\frac{3}{2}}}{24} + \frac{5 a ^{6}}{32} (2 arcsin (\frac{x}{a}) + sin (2 arcsin (\frac{x}{a})))

Add a constant to the solution

= \frac{1}{6} x (a^{2} - x^{2})^{\frac{5}{2}} + \frac{5 a ^{2} x ( - x ^{2} + a ^{2} ) ^{\frac{3}{2}}}{24} + \frac{5 a ^{6}}{32} (2 arcsin (\frac{x}{a}) + sin (2 arcsin (\frac{x}{a}))) + C

\int 4 x^{3} - 10 x d x

\int \frac{4 t ^{3} - t ^{2} + 16 t}{t ^{2} + 4} d t

n = 1 \sum \infty \frac{n}{n ^{2} + 3}

\int (x y^{2} - cos (x) sin (x)) d x

y^{^{'}} - 2 x y = 8 x