$\int 2n \cdot e^{n^2} \, dn$
Solution:
$I = \int 2n \cdot e^{n^2} \, dn$
Let $y = e^{n^2}$
Answer (1)
To solve the integral ∫ 2 n ⋅ e n 2 d n , we can use a technique called substitution, which is a common method for solving integrals in calculus.
Substitution:
We notice that the expression inside the exponential function, n 2 , can be differentiated easily. So, let's set u = n 2 . This means the derivative of u with respect to n is d n d u = 2 n , or d u = 2 n d n .
Rewrite the integral:
Our integral now becomes
∫ 2 n ⋅ e n 2 d n = ∫ e u d u .
This change simplifies the integral significantly because the integral of e u is straightforward.
Solve the integral:
The integral of e u with respect to u is:
∫ e u d u = e u + C , where C is the constant of integration.
Back-substitute:
Finally, substitute back u = n 2 into the expression:
∫ 2 n ⋅ e n 2 d n = e n 2 + C .
Thus, the solution to the integral ∫ 2 n ⋅ e n 2 d n is e n 2 + C , where C is the constant of integration. Using substitution, we have transformed a potentially complex problem into a straightforward one.
