**Evaluate the integral. 2 1 t3 t2 − 1 dt √2 Part 1 of 7…**

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## Question “Evaluate the integral. 2 1 t3 t2 − 1 dt √2 Part 1 of 7…”

Evaluate the integral.

2 |
| ||||

√2 |

Part 1 of 7

Since

2 |
| ||||

√2 |

contains the expression

*t*^{2} − 1,

we will make the substitution

*t* = sec *θ*.

With this, we get*dt* =

$$sec(θ)tan(θ)

Part 2 of 7

Using

*t* = sec *θ*,

we can also say that

| = |
| ||||

= | tanθ. |

Part 3 of 7

Using

*t* = sec *θ*,

we must also convert the integration limits.

If

*t* = 2,

then

θ | = |
sec^{−1}(2) |

= | $$π3 |

and if *t* =

2 |

, then

θ | = |
sec^{−1}
| ||||

= | $$π4 |

Part 4 of 7

Using

*t* = sec *θ*,

we get

2 |
| ||||

√2 |

=

π/3 |
sec | ||

π/4 |

.This can be further simplified to

π/3 |
| ||

π/4 |

,or simply

π/3 |
cos^{2}θ dθ. |

π/4 |

Part 5 of 7

Since we have an even power of cos *x*, to evaluate

π/3 |
cos^{2}θ dθ |

π/4 |

, we must now make the further

substitution.cos^{2}*θ* =

1 |

2 |

1

+ cos

2

*θ*

Part 6 of 7

We have

| = |
sin 2
. |

Part 7 of 7

Now,

sin 2
| = |
+
sin
−
+
sin
. |

After simplifying we have

2 |
| ||||

√2 |

=

## Answer

## Conclusion

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