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प्रश्न
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
उत्तर
Let I = `int t^2/(t + 1)^2*dt`
= `int ((t^3 + 1) - 1)/(t + 1)^2*dt`
= `int ((t + 1)(t^2 - t + 1) - 1)/(t + 1)^2*dt`
= `int [(t^2 - t + 1)/(t + 1) - (1)/((t + 1^2))]*dt`
= `int [((t + 1)(t - 2) + 3)/(t + 1) - (1)/((t + 1)^2)]*dt`
= `int[t - 2 + 3/(t + 1) - 1/((t + 1)^2)]*dt`
= `int t*dt - 2 int 1*dt + 3 int (1)/(t + 1)*dt - int (1)/((t + 1)^2)*dt`
= `t^2/(2) - 2t + 3|log|t + 1| - ((t + 1)-1)/((-1)) + c`
= `t^2/(2) - 2t + 3log|t + 1| + 1/(t + 1) + c`.
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