Use integration by parts to evaluate ∫ 1
3
​
​
tan −1
( x
1
​
)dx (b) Apply the trigonometric substitution to compute the integral ∫ 2
​
2
​
x 3
x 2
−1
​
​
(a) Evaluate the integral ∫ −1/2
1/2
​
1−x 2
1
​
dx using hyperbolic substitutions. (b) Evaluate the integral ∫ 0
6
π
​
​
tan 3
(8x)sec 3
(8x)dx (a) Apply the partial fraction decomposition to evaluate the integral ∫ (x−1)(x 2
+3) 2
x+2
​
dx (b) Sketch the region enclosed by the straight lines 5x−3y+7=0, 2x−5y−1=0, and 3x+2y−11=0. Hence find the area of the region bounded by them.