Question

Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. $$\int\left(a^{2}-x^{2}\right)^{3 / 2} d x$$

Short answer

Question: Find the indefinite integral of \((a^2-x^2)^{3/2}\)dx where a is a positive real number. Answer: The indefinite integral of \((a^2-x^2)^{3/2}\)dx is given by the expression: $$a^4\left(\frac{1}{4}\arcsin\left(\frac{x}{a}\right)+\frac{1}{2}\sin\left(2\arcsin\left(\frac{x}{a}\right)\right)+\frac{1}{8}\left(\arcsin\left(\frac{x}{a}\right)+\frac{1}{2}\sin\left(4\arcsin\left(\frac{x}{a}\right)\right)\right)\right) + C$$

Step-by-step solution

Choose a substitution

Let us choose the substitution \(x=a\sin(u)\), where u is a new variable. This substitution is chosen because it will simplify the square root located inside the expression.

Calculate the derivative of the substitution

Now we need to find the derivative of our substitution with respect to the new variable u, so we can perform a change of variables in the integral. The derivative is: $$\frac{dx}{du}=a\cos(u)$$

Rewrite the integral using the substitution

Let us rewrite the integral using our substitution \(x=a\sin(u)\) and its derivative \(dx=a\cos(u) du\). Replacing x in the original integral: $$\int\left(a^{2}-x^{2}\right)^{3 / 2} d x = \int\left(a^2-a^2\sin^2(u)\right)^{3/2}a\cos(u)du$$

Simplify the integrand

Next, we need to simplify the expression inside the integral: $$\int\left(a^2\left(1-\sin^2(u)\right)\right)^{3/2}a\cos(u)du$$ Recall that \(\cos^2(u) = 1-\sin^2(u)\). Thus, we can replace the expression inside the parentheses with \(\cos^2(u)\): $$\int a^3\cos^3(u)\cdot a\cos(u)du$$

Rewrite the integral as a single power of cosine

Rewrite the integral as a single power of the cosine function: $$a^4 \int \cos^4(u)du$$

Use the double angle identity for cosine

To proceed, we need to use the double angle identity for cosine, which states that: $$\cos^2(x)=\frac{1+\cos(2x)}{2}$$ Now square the identity for cosine and replace \(\cos^4(u)\) with its equivalent expression: $$a^4\int\left(\frac{1+\cos(2u)}{2}\right)^2 du$$

Expand and integrate

Expand the expression inside the integral and integrate each term separately: $$a^4\int\left(\frac{1}{4}+\frac{1}{2}\cos(2u)+\frac{1}{4}\cos^2(2u)\right) du$$ Now integrating each component, we get: $$\frac{1}{4}a^4(u+\frac{1}{2}\sin(2u))+\frac{1}{4}a^4\int\cos^2(2u)du$$

Apply the double angle identity again and integrate

Use the double angle identity again for the last term, replacing \(\cos^2(2u)\) with its equivalent expression and integrate: $$a^4\left(\frac{1}{4}u+\frac{1}{2}\sin(2u)+\frac{1}{8}\left(u+\frac{1}{2}\sin(4u)\right)\right) + C$$ where C is the integration constant.

Substitute back the original variable and simplify

Finally, we need to substitute back our original variable x using the substitution \(x=a\sin(u)\). To find the inverse substitution, we use: $$u=\arcsin\left(\frac{x}{a}\right)$$ Now, replace u in the expression: $$a^4\left(\frac{1}{4}\arcsin\left(\frac{x}{a}\right)+\frac{1}{2}\sin\left(2\arcsin\left(\frac{x}{a}\right)\right)+\frac{1}{8}\left(\arcsin\left(\frac{x}{a}\right)+\frac{1}{2}\sin\left(4\arcsin\left(\frac{x}{a}\right)\right)\right)\right) + C$$ This is the indefinite integral of the given expression.