Approximate only by infinity Series:

Int(exp(-1-1/v)*(1-exp(v^2/(2*t^2)))/v^2, v = 0 .. 1) = 1/exp(2)-Sum(2^(-j)*t^(-2*j)*GAMMA(-2*j+1, 1)*exp(-1)/factorial(j), j = 0 .. infinity);

1/exp(2.)-evalf[10](Sum(eval(2^(-j)*t^(-2*j)*GAMMA(-2*j+1, 1)*exp(-1)/factorial(j), t = 1), j = 0 .. infinity));

#-0.317747611e-1

int(eval(exp(-1-1/v)*(1-exp(v^2/(2*t^2)))/v^2, t = 1), v = 0 .. 1, numeric);

#-0.3177476104e-1

It not full answer.My answer is symbolic not numeric.

dsolve command this type equation can't solve.

In Maple exist another build-in command to solve  integral equation intsolve,but returns unevaluated for me(an error messages).

Download intsolve.mw

Use a dsolve build-in function to solve ODE's:

dsolve(odetemp = 0, U(z));

#U(z) = (1/2)*(2*tan(sqrt(_C1*c)*(_C2+z)*sqrt(2)/(2*c))*sqrt(_C1*c)+sqrt(2)*v)*sqrt(2)

If dsolve can't solve, the only hope is numeric integration the ODE's or another methods(e.g: series,...).

dsolve([odetemp = 0,ics,bcs], numeric);

maplerpimes_v2.mw

#It's a Bug in method = lookup, use another method like:"FTOC" or "FTOCMS"

int(exp(-t)/(1-t), t = 0 .. infinity, CauchyPrincipalValue = true, method = FTOC);

#Ei(1)*exp(-1)

You can see what method uses the int,execute code:

restart:

infolevel[IntegrationTools] := 3;

int(exp(-t)/(1-t), t = 0 .. infinity, CauchyPrincipalValue = true);

Maple simply does not know how to  find closed form of the SUM.

Limit needs this solution from sum,but return unevaluated and for that  this limit evaluation is somehow broken.

Try this only approx:

f := proc (x, n) options operator, arrow; evalf(x*ln(n)-(1/2)*Pi-add(arctan(x/k), k = 1 .. n)) end proc;
x := 2; INF := 10^6; [f(x, INF), evalf(argument(GAMMA(I*x)))];

#[-1.441148036, -1.441150010]

EDITED 03.08.2018:

Download limit.mw

Maple is weak solving inequalities and this case give a incorrect answer(probably a bug).

See attached file:
Download Solve.mw

You can compute numerically this integeral if you assume for constant p:

p := 10;
KK := int(-87.20805639*csgn((p^2+16.703569+(10.0-q)^2+8.174*p)/sqrt((p^2+16.703569+(10.0-q)^2)^2-66.814276*p^2))*(2.*(p^2+16.703569+(10.0-q)^2)^2+66.814276*p^2)/(sqrt((p^2+16.703569+(10.0-q)^2)^2-66.814276*p^2)*((p^2+16.703569+(10.0-q)^2)^4-133.628552*(p^2+16.703569+(10.0-q)^2)^2*p^2+4464.147477*p^4)), q = 0 .. infinity, numeric);

#KK := -0.5804485609e-2

Or Plotting:

func := proc (p) options operator, arrow; evalf(Int((-1)*87.20805639*csgn((p^2+16.703569+(10.0-q)^2+8.174*p)/sqrt((p^2+16.703569+(10.0-q)^2)^2+(-1)*66.814276*p^2))*(2.*(p^2+16.703569+(10.0-q)^2)^2+66.814276*p^2)/(sqrt((p^2+16.703569+(10.0-q)^2)^2+(-1)*66.814276*p^2)*((p^2+16.703569+(10.0-q)^2)^4+(-1)*133.628552*(p^2+16.703569+(10.0-q)^2)^2*p^2+4464.147477*p^4)), q = 0 .. infinity, method = _d01amc)) end proc;
plot([seq([p, func(p)], p = -2 .. 2, 1/10)]);

combine(log(x)+log(y)) assuming(x > 0, y > 0);
#ln(x*y)

combine(log(x)+log(y)) assuming(x > 0);
#ln(x*y)

combine(log(x)+log(y)) assuming(y > 0);
#ln(x*y)

restart;

de := sin(1)*(diff(y(x), x$2))+(1+cos(1)*x^2)*y(x) = -1;

cond := y(-1) = 0, y(1) = 0;

sol := dsolve({cond, de}, numeric);

plots:-odeplot(sol, [x, y(x)]);

It is  not so easy.Only hope is numeric, because dsolve must find integral(probably closed form not exist)  and solve a Transcendental equation(Such equations often do not have closed form solution ) with WhittakerM function.

We can try to find a general solution.

value(dsolve(de));

but Maple can't find  integral and returns unevaluated.

Borrowing code from user https://www.mapleprimes.com/users/tomleslie and adding method _d01ajc to integrate(changing limits to [0,1]) we can speed up code at factor 40,and takes about 2.828 secs on my machine (see the attached).

sumInt-SPEED.mw

Error messages says: "system must be entered as a set/list of expressions/equations"

sys := ((D@@2)(theta1))(t) = t, ((D@@2)(theta2))(t) = t;#this is a list of equations.

# I assume a system of equations, because you did not give any.

sol1 := dsolve({sys, theta1(0) = (1/2)*Pi, theta2(0) = (1/4)*Pi, (D(theta1))(0) = 0, (D(theta2))(0) = 0}, {theta1(t), theta2(t)}, type = numeric, output = listprocedure);

sol1(0);

Workaround is  changing  integration method to _d01ajc that gives 7 correct digits.

Download integral_with_floor.mw

y := func^2; eval(diff(y, func), func = diff(tau(t), t));

#2*(diff(tau(t), t))

Check yours syntax,because is wrong!

Execute in command line for more information about itegration:  ?int

Probably you want as a example:

int(x*y, [x = 4 .. 16, y = 0 .. 2]);

#240

Wrong address.Your question is pure math. Ask that in math forums.

Laplace transform of yours examples not exist. This is only my opinion.

Not sure what is expected from that input.

A correct syntax is:

L1 := inttrans:-laplace(psi1(t)*(diff(z1(t), t)), t, s);
L2 := inttrans:-laplace((diff(psi1(t), t))^2, t, s);

#Return unevaluated