Question

Evaluate the integral. (Use C for the constant of integration.)
(sec2(t) i + t(t2 + 1)2 j + t5 ln(t) k) dt

Answer 1

After applying integration the final answer will be tanx+ 1/6 *(t^2+1)^3+1/6 ln(t)/t^6 - 1/36t^6 +c

What is integration?

Integration is the process of combining smaller parts or data contained in various subsystems into a single operational entity.

Why we need to study integration?

We frequently understand the relationship between two variables' rates of change, but sometimes we also need to understand their direct relationship.

Apply linearity:

$\int\sec^{2}t \, dt +\int\ {t(t^{2}+1} )^{2} } \, dt +\int\ {t^{5}ln(t)} } \, dt$

First

$\int\ {sec^{2}t } \, dt= tanx$

This is standard integral.

Second can be done by substitution method:

Let u= $t^{2}+1$

du= 2t dt

1/2 du= t dt

So

$\int\ {t(t^{2} +1)^{2} } \, dt=\frac{1}{2} \int\ {u^{2} } \, du\\ =\frac{1}{2} \frac{u^{3} }{3}\\ =\frac{u^{3} }{6}+c$

$=\frac{(t^{2}+1) ^{3}}{6} +c$

Now for third using integration by parts:

Let u=ln(t), du=1/t

dv=$t^{5}$ , v= 1/6 $t^{6}$

$\int\ {t^{5} ln(t)} \, dt =\frac{1}{6}\frac{ln(t)}{t^{6} } -\frac{1}{36}t^{6}+c$

To learn more about integration visit the link:

https://brainly.com/question/27419605

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