Question

Evaluate the following integrals. Include absolute values only when needed. intfrac{ln^2x+2ln x-1}{x}dx

Integrals
ANSWERED
asked 2021-01-13
Evaluate the following integrals. Include absolute values only when needed.
\(\int\frac{\ln^2x+2\ln x-1}{x}dx\)

Answers (1)

2021-01-14
\(\text{Given integtral, }\int\frac{\ln^2x+2\ln x-1}{x}dx\)
\(\text{we have to find the given integral.}\)
\(\text{here,}\int\frac{\ln^2x+2\ln x-1}{x}dx\)
\(\text{substitute } \ln x=t\Rightarrow\frac{1}{x}\frac{dx}{dt}=1\Rightarrow\frac{1}{x}dx=dt\)
\(\Rightarrow, \int\frac{\ln^2x+2\ln x-1}{x}dx=\int(t^2+2t-1)dt\)
\(=\int t^2dt+2\int tdt-\int 1dt\)
\(=\frac{t^3}{3}+2\frac{t^2}{2}-t+C\)
\(=\frac{t^3}{3}+t^2-t+C\)
\(\text{undo substitution }t=\ln x\)
\(\Rightarrow\int\frac{\ln^2x+2\ln x-1}{x}dx=\frac{\ln^3x}{3}+\ln^2x-\ln x+C\)
\(\text{this is the required answer}\)
0
 
Best answer

expert advice

Need a better answer?

Relevant Questions

asked 2021-02-22
Evaluate the following integrals. Include absolute values only when needed.
\(\int 3^{-2x}dx\)
asked 2021-06-10
Evaluate the integral \(\int \arctan(x)dx\)
A)\(-\arctan(x)+\ln(x^2+1)+C\)
B)\(\frac{1}{x^2+1}+C\)
C)\(x \arctan(x)-\frac{1}{2}\ln(x^2+1)+C\)
D) \(\ln|\arccos(x)|+C\)
E) \(x\arctan(x)+C\)
asked 2021-05-14
Use the Table of Integrals to evaluate the integral. (Use C for the constant of integration.)
\(\int37e^{74x}\arctan(e^{37x})dx\)
Inverse Trigonometric Forms (92): \(\int u\tan^{-1}u\ du=\frac{u^{2}+1}{2}\tan^{-1}u-\frac{u}{2}+C\)
asked 2021-05-18
Determine whether the following integrals are convergen:
\(\int_5^\infty\frac{\arctan x}{x^2+3x+5}dx\)
asked 2021-05-02
Evaluate the integral. (Use C for the constant of integration.)
\(\int \ln (\sqrt x)dx\)
asked 2021-05-27
Evaluate the indefinite integral as a power series.
\(\int \frac{\tan^{-1}x}{x}dx\)
\(f(x)=C+\sum_{n=0}^\infty\left( \dots \right)\)
What is the radius of convergence R?
asked 2021-06-04
Evaluate the indefinite integral as an infinite series. \(\int \frac{\cos x-1}{x} dx\)
asked 2021-05-04
Evaluate the definite integral.
\(\int_0^1 \sqrt[3]{1+7x}dx\)
asked 2020-11-20
Evaluate the following integrals.
\(\int\frac{4^{\cot x}}{\sin^2x}dx\)
asked 2020-11-08
Evaluate the following integrals.
\(\int\frac{e^x+e^{-x}}{e^x-e^{-x}}dx\)
...