Solve part c step by step.

Solve part c step by step. (c) 1 + COSX Integrate : Integrate: x +3X+2 cos X COS X —cosx dx Evaluate the integral 3cos2x - l)cosx loge x (a) The value of the integral dx is:

\int \frac{x^{3} + 3 x + 2}{{(x^{2} + 1)}^{2} (x + 1)} dx Performing partial fraction \frac{x^{3} + 3 x + 2}{{(x^{2} + 1)}^{2} (x + 1)} = \frac{Bx + A}{(x^{2} + 1)} + \frac{Dx + C}{{(x^{2} + 1)}^{2}} + \frac{E}{x + 1} x^{3} + 3 x + 2 = (Bx + A) (x^{2} + 1) (x + 1) + (Dx + C) (x + 1) + E {(x^{2} + 1)}^{2} x^{3} + 3 x + 2 = (Bx + A) (x^{3} + x^{2} + x + 1) + {Dx}^{2} + Dx + Cx + C + {Ex}^{4} + E + 2 {Ex}^{2} x^{3} + 3 x + 2 = {Bx}^{4} + {Bx}^{3} + {Bx}^{2} + Bx + {Ax}^{3} + {Ax}^{2} + Ax + A + {Dx}^{2} + Dx + Cx + C + {Ex}^{4} + E + 2 {Ex}^{2} x^{3} + 3 x + 2 = (B + E) x^{4} + (B + A) x^{3} + (B + A + D + 2 E) x^{2} + (B + A + D + C) x + A + C + E On comparing coeff . we get B + E = 0 B + A = 1 B + A + D + 2 E = 0 A + B + C + D = 3 A + C + E = 2 On solving we get A = \frac{1}{2}, B = \frac{1}{2}, C = 2, D = 0, E = \frac{- 1}{2} So, \int \frac{x^{3} + 3 x + 2}{{(x^{2} + 1)}^{2} (x + 1)} dx = \frac{1}{2} \int \frac{x + 1}{(x^{2} + 1)} dx + 2 \int \frac{1}{{(x^{2} + 1)}^{2}} dx - \frac{1}{2} \int \frac{1}{x + 1} dx Now consider \int \frac{x + 1}{(x^{2} + 1)} dx = \int \frac{x}{x^{2} + 1} dx + \int \frac{1}{x^{2} + 1} dx = \frac{\ln (x^{2} + 1)}{2} + \tan^{- 1} x + c_{1} Now consider \int \frac{1}{{(x^{2} + 1)}^{2}} dx Apply reduction formula \int \frac{1}{{({ax}^{2} + b)}^{n}} dx = \frac{2 n - 3}{2 b (n - 1)} \int \frac{1}{{({ax}^{2} + b)}^{n - 1}} dx + \frac{x}{2 b (n - 1) {({ax}^{2} + b)}^{n - 1}} With a = 1, b = 1 and n = 2 = \frac{x}{2 (x^{2} + 1)} + \frac{1}{2} \int \frac{1}{x^{2} + 1} dx = \frac{x}{2 (x^{2} + 1)} + \frac{\tan^{- 1} x}{2} + c_{2} So, \int \frac{x^{3} + 3 x + 2}{{(x^{2} + 1)}^{2} (x + 1)} dx = \frac{1}{2} \int \frac{x + 1}{(x^{2} + 1)} dx + 2 \int \frac{1}{{(x^{2} + 1)}^{2}} dx - \frac{1}{2} \int \frac{1}{x + 1} dx = \frac{\ln (x^{2} + 1)}{4} + \frac{3 \tan^{- 1} x}{2} + \frac{x}{(x^{2} + 1)} - \frac{\ln (x + 1)}{2} + C (where C = c_{1} + c_{2})

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Solve part c step by step. Solve part c step by step. (c) 1 + COSX Integrate : Integrate: x +3X+2 cos X COS X —cosx dx Evaluate the integral 3cos2x - l)cosx loge x (a) The value of the integral dx is:

Solve part c step by step.

Solve part c step by step. (c) 1 + COSX Integrate : Integrate: x +3X+2 cos X COS X —cosx dx Evaluate the integral 3cos2x - l)cosx loge x (a) The value of the integral dx is: