Problem 16 - Essential Calculus

Question

Use integration tables to find the integral. $$ \int \sqrt{x} \arctan x^{3 / 2} d x $$

Step-by-Step Solution

Answer

The integral \( \int \sqrt{x} \arctan x^{3 / 2} dx = \arctan x^{3 / 2} * 2/3 x^{3 / 2} - 2/3 \arctan x^{3 / 2} + c\)

Step 1: Identify the parts for integration by parts

Integration by parts is a technique of integration, which is useful when the integral is a product of two functions, one of which is easily integrated. The integral \( \int \sqrt{x} \arctan x^{3 / 2} dx \) appears to be a product of \( \arctan x^{3 / 2} \) and \( \sqrt{x} \). The method requires us to identify two parts in the integral: a part that can be easily integrated which we will call 'dv', and a part when differentiated becomes simpler which we will call 'u'. Here, it can be seen that \(dv = \sqrt{x}dx\) and \(u = \arctan x^{3 / 2} \).

Step 2: Calculate du and V

We will now calculate du which is the derivative of u and V which is the integral of dv. From \(u = \arctan x^{3 / 2}\), we differentiate u to get \(du = 1/(1+(x^{3 / 2})^2) * 3 / 2 \sqrt{x} dx\). Taking the integral of dv = \( \sqrt{x}dx \) will yield \(V = 2/3 x^{3 / 2}\).

Step 3: Apply the integration by parts formula

Now we can apply the integration by parts formula: \( \int u dv = uV - \int V du\). This will give us \( \int \sqrt{x} \arctan x^{3 / 2} dx = \arctan x^{3 / 2} * 2/3 x^{3 / 2} - \int 2/3 x^{3 / 2} * 1/(1+(x^{3 / 2})^2)* 3 / 2 \sqrt{x} dx\).

Step 4: Simplify the integral

The integral can be simplified to \( \int x^{3 / 2}/(1+(x^{3 / 2})^2) dx\). Now apply the substitution \( t = x^{3 / 2} \) to make \( dt = 3/2 \sqrt{x} dx \), or equivalently \( dx = 2/3 dt/\sqrt{t} \). This makes the new integral \(2/3 \int dt/(1+t^2) \). The integral of 1/(1 + t^2) is a standard integral and it is \(\arctan t\). Therefore, this term integrates to \(2/3 \arctan t + c\), where c is the constant of integration.

Step 5: Substitute back for t

Remember that we let \(t = x^{3 / 2}\). Substituting back gives \(2/3 \arctan x^{3 / 2} + c\). This gives the final answer: \( \int \sqrt{x} \arctan x^{3 / 2} dx = \arctan x^{3 / 2} * 2/3 x^{3 / 2} - 2/3 \arctan x^{3 / 2} + c\)