Question
Evaluate the integral
2echg2
Evaluate
∫0+∞4te−t2dt×cheg×g
Evaluate the integral
More Steps

Use function domain and discontinuity points to transform the expression with the formula ∫acf(x)dx=∫abf(x)dx+∫bcf(x)dx
∫0+∞4te−t2dt
By definition,rewrite the improper integral using one-sided limit and a definite integral
a→+∞lim(∫0a4te−t2dt)
Evaluate the integral
More Steps

Evaluate
∫0a4te−t2dt
Evaluate the integral
∫4te−t2dt
Use the property of integral ∫kf(x)dx=k∫f(x)dx
4×∫te−t2dt
Use the substitution dt=−2t1dv to transform the integral
4×∫te−t2(−2t1)dv
Simplify
4×∫−2e−t2dv
Use the substitution v=−t2 to transform the integral
4×∫2−evdv
Simplify the expression
4×∫−2evdv
Rewrite the expression
4×∫−21evdv
Use the property of integral ∫kf(x)dx=k∫f(x)dx
4(−21)×∫evdv
Multiply the numbers
−2×∫evdv
Use the property of integral ∫exdx=ex
−2ev
Substitute back
−2e−t2
Return the limits
(−2e−t2)0a
Substitute the values into formula
−2e−a2−(−2e−02)
Calculate
−2e−a2−(−2e−0)
Calculate
−2e−a2−(−2e0)
Evaluate the power
−2e−a2−(−2×1)
Any expression multiplied by 1 remains the same
−2e−a2−(−2)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
−2e−a2+2
a→+∞lim(−2e−a2+2)
Rewrite the expression
a→+∞lim(−2e−a2)+a→+∞lim(2)
Calculate
More Steps

Evaluate
a→+∞lim(−2e−a2)
Rewrite the expression
−2×a→+∞lim(e−a2)
Calculate
−2×0
Any expression multiplied by 0 equals 0
0
0+a→+∞lim(2)
Calculate
0+2
Calculate
2
2cheg×g
Multiply the terms
2cheg2
Solution
2echg2
Show Solution
