Evaluate the indefinite integral `int (csc(t)cot(t))/(1+9csc^2(t))dt`
Expert Answers
| Certified Educator
calendarEducator since 2012
write1,284 answers
starTop subjects are Math and Science
`int (csc(t) cot(t) ) / (1+9csc^2(t)) dt`
`= int (csc(t)cot(t))/(1+(3csct)^2) dt`
Note that the derivative of co-secant function is (csc u)' = -csc u cot u. Since the numerator of the integrand is similar to the of the derivative of co-secant, use u-substitution method to integrate.
So let,
`u = 3csc(t)`
Then, differentiate u.
`du = -3csc(t)cot(t) dt`
`-(du)/3=csc(t)cot(t) dt`
And replace the t variable of the integrand with u.
`int 1/(1+u^2)*(- (du)/3)`
`=-1/3 int 1/(1+u^2) du`
Apply the formula `int 1/(a^2 + u^2)du = 1/a tan^(-1) u/a+C` .
`= -1/3 tan^(-1) u + C`
Substitute back u=3csc(t) to return to the original variable.
`=-1/3 tan^(-1) (3csc (t) ) + C`
Hence, `int (csc(t)cot(t))/(1+9csc^2t)dt = -1/3 tan^(-1) (3csc(t))+C` .
Related Questions
- `int 5^t sin(5^t) dt` Evaluate the indefinite integral.
- 1 Educator Answer
- `int (dt)/(t^2 sqrt(t^2 - 16))` Evaluate the integral
- 2 Educator Answers
- `int (csc^2(t))/cot(t) dt` Find the indefinite integral.
- 1 Educator Answer
- `int sqrt(cot(x)) csc^2(x) dx` Evaluate the indefinite integral.
- 1 Educator Answer
- `int t sec^2 2t dt` Evaluate the integral
- 1 Educator Answer