Evaluate.
$\int_{1}^{e^{3}} (3 x + \frac{1}{x}) d x$
$\int_{1}^{e^{3}} (3 x + \frac{1}{x}) d x = ◻$ (Type an exact answer in terms of e.)

Answer

Expert–verified

Hide Steps

Answer

Add the results of the two integrals to get the final answer: $\frac{3}{2} + \frac{3 e^{6}}{2}$ .

Steps

Step 1 ：Split the integral into two separate integrals: $\int_{1}^{e^{3}} 3 x d x$ and $\int_{1}^{e^{3}} \frac{1}{x} d x$ .

Step 2 ：Solve the first integral using the power rule to get the antiderivative $\frac{3 x^{2}}{2}$ .

Step 3 ：Solve the second integral using the logarithmic rule to get the antiderivative $\ln (x)$ .

Step 4 ：Substitute the upper and lower limits of the integral into the antiderivatives to get $\frac{3 e^{6}}{2} - \frac{3}{2}$ and $3 \ln (e^{3}) - \ln (1)$ .

Step 5 ：Simplify the results to get $\frac{3 e^{6}}{2} - \frac{3}{2}$ and $3 \cdot 3 - 0$ .

Step 6 ：Add the results of the two integrals to get the final answer: $\frac{3}{2} + \frac{3 e^{6}}{2}$ .

Evaluate.∫1e3(3x+1x)dx∫1e3(3x+1x)dx=◻ (Type an exact answer in terms of e.)

Answer

Popular Problems

Evaluate.
$\int_{1}^{e^{3}} (3 x + \frac{1}{x}) d x$
$\int_{1}^{e^{3}} (3 x + \frac{1}{x}) d x = ◻$ (Type an exact answer in terms of e.)