Question

1. -/1.07 points evaluate the integral. (use c for the constant of integration.) $int t^{6}ln(t)dt$ resources watch it

Explanation:

Step1: Use integration - by - parts formula

The integration - by - parts formula is $\int u\mathrm{d}v=uv-\int v\mathrm{d}u$. Let $u = \ln(t)$ and $\mathrm{d}v=t^{6}\mathrm{d}t$. Then $\mathrm{d}u=\frac{1}{t}\mathrm{d}t$ and $v=\frac{t^{7}}{7}$.

Step2: Apply the formula

$\int t^{6}\ln(t)\mathrm{d}t=\frac{t^{7}}{7}\ln(t)-\int\frac{t^{7}}{7}\cdot\frac{1}{t}\mathrm{d}t$.

Step3: Simplify the second integral

$\int\frac{t^{7}}{7}\cdot\frac{1}{t}\mathrm{d}t=\frac{1}{7}\int t^{6}\mathrm{d}t$.

Step4: Integrate $t^{6}$

$\frac{1}{7}\int t^{6}\mathrm{d}t=\frac{1}{7}\cdot\frac{t^{7}}{7}+C=\frac{t^{7}}{49}+C$.

Step5: Get the final result

$\int t^{6}\ln(t)\mathrm{d}t=\frac{t^{7}}{7}\ln(t)-\frac{t^{7}}{49}+C$.

Answer:

$\frac{t^{7}}{7}\ln(t)-\frac{t^{7}}{49}+C$