Evalute the following indefinite integrals:
1. \(\int sec^2xdx/tan^2x+3tanx+2 \)
2.\(\int x^4+x^2-1/(x^3+x) *dx\)
3. \(\int e^{2x}/\sqrt[3]{e^x-1}*dx\)
NO. 1
Take the integral:
integral (sec^2(x))/(tan^2(x)+3 tan(x)+2) dx
For the integrand (sec^2(x))/(tan^2(x)+3 tan(x)+2), substitute u = tan(x) and du = sec^2(x) dx:
= integral 1/(u^2+3 u+2) du
For the integrand 1/(u^2+3 u+2), complete the square:
= integral 1/((u+3/2)^2-1/4) du
For the integrand 1/((u+3/2)^2-1/4), substitute s = u+3/2 and ds = du:
= integral 1/(s^2-1/4) ds
Factor -1/4 from the denominator:
= integral 4/(4 s^2-1) ds
Factor out constants:
= 4 integral 1/(4 s^2-1) ds
Factor -1 from the denominator:
= -4 integral 1/(1-4 s^2) ds
For the integrand 1/(1-4 s^2), substitute p = 2 s and dp = 2 ds:
= -2 integral 1/(1-p^2) dp
The integral of 1/(1-p^2) is tanh^(-1)(p):
= -2 tanh^(-1)(p)+constant
Substitute back for p = 2 s:
= -2 tanh^(-1)(2 s)+constant
Substitute back for s = u+3/2:
= -2 tanh^(-1)(2 u+3)+constant
Substitute back for u = tan(x):
= -2 tanh^(-1)(2 tan(x)+3)+constant
Which is equivalent for restricted x values to:
Answer: | = log(sin(x)+cos(x))-log(sin(x)+2 cos(x))+constant
NO. 2
integral (x^4-1/(x^3+x)+x^2) dx
Integrate the sum term by term and factor out constants:
= - integral 1/(x^3+x) dx+ integral x^4 dx+ integral x^2 dx
For the integrand 1/(x^3+x), use partial fractions:
= - integral (1/x-x/(x^2+1)) dx+ integral x^4 dx+ integral x^2 dx
Integrate the sum term by term and factor out constants:
= integral x/(x^2+1) dx- integral 1/x dx+ integral x^4 dx+ integral x^2 dx
For the integrand x/(x^2+1), substitute u = x^2+1 and du = 2 x dx:
= 1/2 integral 1/u du- integral 1/x dx+ integral x^4 dx+ integral x^2 dx
The integral of 1/u is log(u):
= (log(u))/2- integral 1/x dx+ integral x^4 dx+ integral x^2 dx
The integral of 1/x is log(x):
= (log(u))/2-log(x)+ integral x^4 dx+ integral x^2 dx
The integral of x^4 is x^5/5:
= x^5/5+(log(u))/2-log(x)+ integral x^2 dx
The integral of x^2 is x^3/3:
= (log(u))/2+x^5/5+x^3/3-log(x)+constant
Substitute back for u = x^2+1:
Answer: | = x^5/5+x^3/3+1/2 log(x^2+1)-log(x)+constant
NO. 3
Take the integral:
integral e^(2 x)/(e^x-1)^(1/3) dx
For the integrand e^(2 x)/(e^x-1)^(1/3), substitute u = e^x and du = e^x dx:
= integral u/(u-1)^(1/3) du
For the integrand u/(u-1)^(1/3), substitute s = u-1 and ds = du:
= integral (s+1)/s^(1/3) ds
Expanding the integrand (s+1)/s^(1/3) gives s^(2/3)+1/s^(1/3):
= integral (s^(2/3)+1/s^(1/3)) ds
Integrate the sum term by term:
= integral s^(2/3) ds+ integral 1/s^(1/3) ds
The integral of s^(2/3) is (3 s^(5/3))/5:
= (3 s^(5/3))/5+ integral 1/s^(1/3) ds
The integral of 1/s^(1/3) is (3 s^(2/3))/2:
= (3 s^(5/3))/5+(3 s^(2/3))/2+constant
Substitute back for s = u-1:
= 3/5 (u-1)^(5/3)+3/2 (u-1)^(2/3)+constant
Substitute back for u = e^x:
= 3/5 (e^x-1)^(5/3)+3/2 (e^x-1)^(2/3)+constant
Which is equal to:
Answer: | = 3/10 (e^x-1)^(2/3) (2 e^x+3)+constant
