-2/√3 atan ((2t + 1)/√3) + C
Explanation:
∫ (t − 1) / (1 − t³) dt
Factor the difference of cubes:
∫ (t − 1) / ((1 − t)(1 + t + t²)) dt
Divide:
∫ -1 / (1 + t + t²) dt
-∫ 1 / (t² + t + 1) dt
Complete the square:
-∫ 1 / (t² + t + ¼ + ¾) dt
-∫ 4 / (4t² + 4t + 1 + 3) dt
-∫ 4 / ((2t + 1)² + 3) dt
If u = 2t + 1, du = 2 dt:
-∫ 2 / (u² + 3) du
Use an integral table, or use trigonometric substitution:
-2 (1/√3) atan (u/√3) + C
-2/√3 atan (u/√3) + C
Substitute back:
-2/√3 atan ((2t + 1)/√3) + C
? what
Explanation: