Evaluate the following integrals
(i) `int_(R)^(oo)(GMm)/(x^(2))dx`
(ii) `int_(r_(1))^(r_(2)) -k(q_1q_2)/(x^(2))dx`
(iii) `int_(u)^(v) Mv dv`
(iv) `int_(0)^(oo) x^(-1//2) dx`
(v) `int_(0)^(pi//2) int sinx dx `
(vi) `int_(0)^(pi//2) int cosx dx`
(vii) `int_(-pi//2)^(pi//2) cos x dx`

Answer

(i) `(GMm)/(R)`
(ii) `kq_1q_2((1)/(r_2)- (1)/(r_1))`
(iii) `(1)/(2) M (v^(2) - u^(2))`
(iv) `oo`
(v) `1`
(vi) `1`
(vii) `2`

Explanation

(i) `overset(oo) underset(R) int (GMm)/(x^(2)) dx = GMm overset(oo) undersetR int x^(-2) dx `
`= GMm[(x^(-1))/(-1)]_(R)^(oo) = (GMm)/(R)`
(ii) `overset(r_2) underset (r_1) int (-kq_1q_2)/(x^(2)) dx `
`= - kq_1 q_2 [(x^(-1))/(-1)] = kq_1q_2[(1)/(r_2)-(1)/(r_1)]`
(iii) `overset(v) underset(u) int Mv dv = M[(v^2)/(2)]_(u)^(v)= (1)/(2) M)(v^(2) -u^(2))`
(iv) `overset(oo) underset (0) int x^(-1//2) dx = [ (x^(1//2))/((1)/(2))] _(0)^(oo) =oo`
(v) `overset(pi//2) underset(0) int sinx dx = [-cosx]_0^(pi//2)= 1`
(vi) `overset(pi//2)underset0 int cos x dx = [sin x] _(0)^(pi//2) = 1 `
(vii) `overset(pi//2) underset (-pi //2) int cosx dx = [sinx]_(pi//2)^(pi//2) =2`