Answer
$\tan ti+\frac{1}{8}(t^2+1)^4j+(\dfrac{1}{3}t^3 \ln t-\dfrac{1}{9}t^3)k+C$
Work Step by Step
Given: $\int (\sec^2 ti+t(t^2+1)^3j+t^2 \ln t k)dt$
In order to evaluate the integral we will have to integrate each component of the function individually.
Let $I=\int (\sec^2 ti+t(t^2+1)^3j+t^2 \ln t k)dt$
Thus,
$I=\int \sec^2 tdti+\int t(t^2+1)^3dtj+\int t^2 \ln t dtk$
$I=\tan ti+ Aj+Bk$ ... (1)
Here, $A=\int t(t^2+1)^3dt$ and $B=\int t^2 \ln t dt$
Consider $A=\int t(t^2+1)^3dt$
suppose $p=t^2+1 \implies dp=2tdt$
Thus, $A=\frac{1}{2}\int p^3dp=\frac{1}{8}p^4=\frac{1}{8}(t^2+1)^4$
Now, consider $B=\int t^2 \ln t dt=\ln t (1/3t^3)-\int (1/3t^3) \dfrac{1}{t} dt=\dfrac{1}{3}t^3 \ln t-\dfrac{1}{9}t^3$
Plug in the values of A and B in equation (1) and we get
$I=\tan ti+\frac{1}{8}(t^2+1)^4j+(\dfrac{1}{3}t^3 \ln t-\dfrac{1}{9}t^3)k+C$
Here, C is the constant of integration.
