EvaluateIntegral (3x-2)*sqrt(x^2+x+1)dx Share with your friends Share 26 Manbar Singh answered this Let I = ∫3x-2x2+x+1 dxLet 3x-2 = A × ddxx2+x+1 + B⇒3x - 2 = A2x + 1 + B⇒3x - 2 = 2Ax + A + BComparing the coefficient of x on both sides, we get2A = 3 ⇒ A = 32Comparing the constants on both sides, we getA + B = -2⇒32+B = -2⇒B = -72Now, 3x-2 = 322x+1-72Now, I = ∫322x+1-72x2+x+1 dx=32∫2x+1x2+x+1 dx - 72∫x2+x+1 dxI=32I1 - 72I2 ........1Now, I1 = ∫2x+1x2+x+1 dxput x2+x+1 = t⇒2x+1dx = dtNow, I1 = ∫t dt = 23t3/2 + C1 = 23x2+x+13/2 + C1Now, I2 = ∫x2+x+1 dx=∫x2+x+14-14+1 dx=∫x + 122 + 322 dx=12x+12x + 122 - 322 + 12×34log x+12+x + 122 - 322 + C2=12x+12x + 122 - 322 +34log x+12+x + 122 - 322 + C2Now, I = x2+x+13/2 - 74x+12x2+x+1 +34log x+12+x2+x+1 + C1 + C2⇒I = x2+x+13/2 -72x+18x2+x+1 - 2116logx + 12 + x2+x+1 + C, where C = C1 + C2 44 View Full Answer
