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Math  /  Calculus

Question8. Integrate: ∫−1t2+11t+tdt\int \frac{-\frac{1}{t^{2}}+1}{\sqrt{\frac{1}{t}+t}} d t

Studdy Solution

STEP 1

What is this asking? We need to find the integral of a function that looks a little scary, but we'll break it down and make it super manageable! Watch out! Don't get overwhelmed by the square root and the fraction inside.
We'll use a *u*-substitution to simplify things!

STEP 2

1. Simplify the integrand
2. Perform *u*-substitution
3. Integrate
4. Substitute back

STEP 3

Let's rewrite the integrand to make it easier to work with.
We can rewrite −1t2-\frac{1}{t^{2}} as −t−2-t^{-2}.
This gives us: ∫−t−2+11t+tdt \int \frac{-t^{-2} + 1}{\sqrt{\frac{1}{t} + t}} dt

STEP 4

Now, let's rewrite the 1t\frac{1}{t} inside the square root as t−1t^{-1}: ∫−t−2+1t−1+tdt \int \frac{-t^{-2} + 1}{\sqrt{t^{-1} + t}} dt

STEP 5

Let's choose our *u*.
A good choice seems to be the stuff inside the square root: u=t−1+tu = t^{-1} + t.
This is because its derivative, dudt=−t−2+1\frac{du}{dt} = -t^{-2} + 1, appears in the numerator!
This is exactly what we want for a *u*-substitution.

STEP 6

Now, let's find *du*.
We have dudt=−t−2+1\frac{du}{dt} = -t^{-2} + 1, so du=(−t−2+1)dtdu = (-t^{-2} + 1) dt.

STEP 7

Let's substitute *u* and *du* into our integral: ∫1udu \int \frac{1}{\sqrt{u}} du This looks much simpler!

STEP 8

We can rewrite the integral as: ∫u−12du \int u^{-\frac{1}{2}} du

STEP 9

Now, we can use the power rule for integration, which says ∫xndx=xn+1n+1+C\int x^n dx = \frac{x^{n+1}}{n+1} + C.
In our case, n=−12n = -\frac{1}{2}, so n+1=−12+1=12n+1 = -\frac{1}{2} + 1 = \frac{1}{2}.

STEP 10

Applying the power rule, we get: ∫u−12du=u1212+C \int u^{-\frac{1}{2}} du = \frac{u^{\frac{1}{2}}}{\frac{1}{2}} + C

STEP 11

Simplifying the result, we have: 2u12+C=2u+C 2u^{\frac{1}{2}} + C = 2\sqrt{u} + C

STEP 12

Finally, let's substitute back the original expression for *u*, which was t−1+t=1t+tt^{-1} + t = \frac{1}{t} + t: 21t+t+C 2\sqrt{\frac{1}{t} + t} + C

STEP 13

The solution to the integral is 21t+t+C2\sqrt{\frac{1}{t} + t} + C.

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