Answer :

Let I =


cos 2x = 2cos2 x – 1 and cos 2α = 2cos2 α – 1


I =


I =


As, a2 – b2 = (a+b)(a-b)


I =


I =


I =


I = 2 sin x + 2 cos α


I = 2(sin x + x cos α) + C


OR


Let, I =


I =


I =


Let, I1 = and I2 =


I = I1 + I2 …(1)


First we will calculate I1 and I2 and then add both to get the answer.


Calculation of I1 :


Let x2 + 2x + 3 = u


du = 2(x + 1)dx


(x + 1)dx = du/2


I1 =


I1 = + C1


I1 = ...(2)


Calculation of I2 :


I2 =


I2 =


Use the formula:


I2 =


I2 = …(3)


from equation 1, 2 and 3 we have –


I =


I =


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