Help with a few single variable questions

[mcl116]

1. Solve for x: e^(3x-1)=5e^(4x)

2. Find an equation for the line tangent to the graph of f(x)= x^2lnx @ x=e^3

3.Evaluate the indefinite integral t^2/(t^3+3)dt

4. Evaluate the definite integral between e^3 & e^5 1/(xlnx)dx

If you could just show the work so I can see the steps and how to get to these answers in the future, that would be great.

Thanks for the help!

 

[galactus]

3.Evaluate the indefinite integral \(\displaystyle \int\frac{t^{2}}{t^{3}+3}dt\)

Make a substitution. Let \(\displaystyle u=t^{3}+3, \;\ du=3t^{2}dt\)

4. Evaluate the definite integral \(\displaystyle \int_{e^{3}}^{e^{5}}\frac{1}{xlnx}dx\)

Make a substitution. Let \(\displaystyle u=ln(x), \;\ du=\frac{1}{x}dx\)

2. Find an equation for the line tangent to the graph of f(x)= x^2lnx @ x=e^3

Find f'(x). Sub in x=e^3 to find the slope, m, at that point. Sub e^3 into f(x) to find y at that point.

Sub everything into y=mx+b and solve for b. Your're finished.

 

[Deleted member 4993]

1. Solve for x: e^(3x-1)=5e^(4x)........Take 'ln' of both sides.

2. Find an equation for the line tangent to the graph of f(x)= x^2lnx @ x=e^3

Slope of the tangent line is df/dx . The line goes through the tangent point.

3.Evaluate the indefinite integral t^2/(t^3+3)dt

4. Evaluate the definite integral between e^3 & e^5 1/(xlnx)dx

If you could just show the work so I can see the steps and how to get to these answers in the future, that would be great.

Thanks for the help!