Evaluate {eq}\displaystyle \int_{1}^{2} \tan^{-1} (t) \ \text{d}t {/eq}

Question:

Definite Integral:

If the boundary values (upper and lower limits) are indicated in the integral symbol, then the integral is classified as definite. The general expression for definite integral is given by

$$\int\limits_b^a {f\left( q \right)} \;\text{d}q $$

where {eq}a {/eq} and {eq}b {/eq} are upper and lower limits, and {eq}f(q) {/eq} is the integrand of the definite integral.

Integration by Parts:

In calculus, integration by parts is a method used to integrate the product of two functions. In this method, we apply the ILATE rule to choose which function comes first and which one comes second. The formula of integration by parts is given by

$$\displaystyle \int {y\left( x \right) \times f\left( x \right)} \;\text{ d}x = y\left( x \right)\int {f\left( x \right)} \text{ d}x - \int {\left( {\dfrac{\text{d}}{{\text{d}x}}y\left( x \right)\int {f\left( x \right)} \text{ d}x} \right)} \text{ d}x, $$

where {eq}y(x) {/eq} and {eq}f(x) {/eq} are the first and second functions, respectively.

ILATE Rule:

The following list helps us to chose the first and second functions when applying integration by parts method.

1. Inverse trigonometric function (Top priority)
2. Logarithmic function
3. Algebraic function
4. Trigonometric function
5. Exponent function (Lowest priority)

Answer and Explanation: