Question

Evaluate the definite integral. $$ \int_{0}^{\pi / 4} \sec ^{2} t \sqrt{\tan t} d t $$

Short answer

The result of the evaluation of the definite integral is \( \frac{2}{3} \).

Step-by-step solution

Set Up the U-Substitution

In cases like these, a common method is to use a u-substitution to simplify the integral. Here, select \( u = \tan(t) \) as it is the inside function. Hence, the derivative of \( u = \tan(t) \) will simplify the \( \sec^{2}(t) \) in the current integral. Compute \( du \) i.e. derivative of \( u = \tan(t) \) which will be \( du = \sec^{2}(t) dt \).

Substitute 'u' in the Integral

After establishing \( du = \sec^{2}(t) dt \), the substitution can be performed. The integral now simplifies to \( \int{(u)^{1/2} d(u)} \) or \( \int{u^{1/2} du} \). The limits of the integral will change. When \( t = 0 \), \( u = \tan(0) = 0 \). So, the lower limit stays at 0. When \( t = \frac{\pi}{4} \), \( u = \tan(\frac{\pi}{4}) = 1 \). So, the upper limit becomes 1.

Compute the Definite Integral

Compute the definite integral \( \int_0^{1}{u^{1/2} du} \), using power rule of integration, which states \( \int{x^{n} dx} = \frac{x^{n+1}}{n+1} \) (for \( n \neq -1 \)). Here, \( n = 1/2 \), so the power rule of integration yields \( \left . \frac{2}{3}u^{3/2} \right |_0^{1} \).

Compute the Result Using the New Limits

Evaluate this expression at \( u = 1 \) which is the upper limit now and subtract from it the expression evaluated at \( u = 0 \), the lower limit. Thus, obtaining \(\left . \frac{2}{3}u^{3/2} \right |_0^{1} = \frac{2}{3}*1^{3/2} - \frac{2}{3}*0^{3/2} = \frac{2}{3} - 0 = \frac{2}{3}\).