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Question: How do you find the antiderivative of \[x\sqrt {100 - {x^2}} dx\] ?...

How do you find the antiderivative of x100−x2dxx\sqrt {100 - {x^2}} dx ?

Explanation

Solution

Hint : We need to evaluate ∫x100−x2dx\int {x\sqrt {100 - {x^2}} dx} . We know that the term inside the integral sign is called integrand. We simplify the integrand by making a substitution that is we put t=100−x2t = 100 - {x^2} . After simplification we apply the integral. In the final we need the solution in terms of ‘x’ only. Here we have an indefinite integral.

Complete step by step solution:
Given,
∫x100−x2dx\int {x\sqrt {100 - {x^2}} dx}
let’s put t=100−x2t = 100 - {x^2}
Differentiating with respect to ‘x’ we have,
dt=−2xdxdt = - 2xdx
Substituting these in the integral we have,
∫x100−x2dx=∫100−x2.xdx\int {x\sqrt {100 - {x^2}} dx} = \int {\sqrt {100 - {x^2}} .xdx}
=∫100−x2.xdx= \int {\sqrt {100 - {x^2}} .xdx}
Multiply and divide by −2 - 2 .
=−12∫100−x2.(−2xdx)= - \dfrac{1}{2}\int {\sqrt {100 - {x^2}} .( - 2xdx)}
Then we have,
=−12∫t.dt= - \dfrac{1}{2}\int {\sqrt t .dt}
=−12∫(t)12.dt= - \dfrac{1}{2}\int {{{\left( t \right)}^{\dfrac{1}{2}}}.dt}
Now applying the integrating we have,
=−12×(t12+1)(12+1)+C= - \dfrac{1}{2} \times \dfrac{{\left( {{t^{\dfrac{1}{2} + 1}}} \right)}}{{\left( {\dfrac{1}{2} + 1} \right)}} + C
where ‘C’ is the integration constant.
=−12×(t32)(32)+C= - \dfrac{1}{2} \times \dfrac{{\left( {{t^{\dfrac{3}{2}}}} \right)}}{{\left( {\dfrac{3}{2}} \right)}} + C
=−22×t323+C= - \dfrac{2}{2} \times \dfrac{{{t^{\dfrac{3}{2}}}}}{3} + C
=−13t32+C= - \dfrac{1}{3}{t^{\dfrac{3}{2}}} + C
But we need the answers in terms of ‘x’ only. We have taken substitution t=100−x2t = 100 - {x^2} .
=−13(100−x2)32+C= - \dfrac{1}{3}{\left( {100 - {x^2}} \right)^{\dfrac{3}{2}}} + C
Thus we have,
∫x100−x2dx==−13(100−x2)32+C\int {x\sqrt {100 - {x^2}} dx} = = - \dfrac{1}{3}{\left( {100 - {x^2}} \right)^{\dfrac{3}{2}}} + C , where ‘C’ is the integration constant.
So, the correct answer is “ ∫x100−x2dx==−13(100−x2)32+C\int {x\sqrt {100 - {x^2}} dx} = = - \dfrac{1}{3}{\left( {100 - {x^2}} \right)^{\dfrac{3}{2}}} + C ”.

Note : Here we have an indefinite integral that is no upper limit and lower limit. Hence, in the case of indefinite integral we have integration constant. In definite integral we have lower limits and upper limits. Hence, in the case of definite integral we don’t have integration constant. As we can see in the above problem by using the substitute rule we can simplify the problem easily.