148k views
4 votes
How do you do this question?

How do you do this question?-example-1
User Aquinas
by
7.2k points

2 Answers

1 vote

Explanation:

∫₁² [e^(1/x⁴) / x⁵] dx

∫₁² [e^(x⁻⁴) x⁻⁵] dx

If u = x⁻⁴, then du = -4x⁻⁵ dx, and -¼ du = x⁻⁵ dx.

∫ e^u (-¼ du)

-¼ ∫ e^u du

-¼ e^u + C

-¼ e^(x⁻⁴) + C

Evaluate between x=1 and x=2.

-¼ e^(2⁻⁴) − -¼ e^(1⁻⁴)

-¼ e^(1/16) + ¼ e

(e − ¹⁶√e) / 4

User Weiweishuo
by
7.9k points
7 votes

Answer:


=\frac{e-\sqrt[16]{e}}{4}

Explanation:

So we have the definite integral:


\int\limits^2_1{\frac{e^{(1)/(x^4)}}{x^5} \, dx

Again, we can use u-substitution. Let u equal 1/x^4. So:


u=(1)/(x^4)=x^(-4)

Find the derivative:


du=-4x^(-5)=-4((1)/(x^5))dx

Divide by -4. So du is:


-(1)/(4)du=(1)/(x^5)dx

Of course, we also need to change our bounds. Substitute 1 and 2 into u:


u=1/(2)^4=1/16\\u=1/(1)^4=1/1=1

Therefore, our new bounds are from 1 to 1/16.

So, make the substitutions:


=-(1)/(4)\int\limits^(1)/(16)_1 {e^u} \, du

The integral of e^u is just e^u. So:


=-(1)/(4)(e^u)

Evaluate for the bounds:


=-(1)/(4)(e^{(1)/(16)}-e^(1))

Simplify:


=-\frac{\sqrt[16]{e}-e}{4}

Distribute:


=\frac{e-\sqrt[16]{e}}{4}

And we're done!

User Igor Shmukler
by
8.5k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories