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9 wrz 2024 · Rule: Integrals of Exponential Functions. Exponential functions can be integrated using the following formulas. ∫exdx = ex + C (5.6.1) ∫axdx = ax ln a + C (5.6.2) Example 5.6.1: Finding an Antiderivative of an Exponential Function. Find the antiderivative of the exponential function e−x. Solution.
Exponential functions can be integrated using the following formulas. ∫ exdx = ex+C ∫ axdx = ax lna +C ∫ e x d x = e x + C ∫ a x d x = a x ln a + C. The nature of the antiderivative of ex e x makes it fairly easy to identify what to choose as u u. If only one e e exists, choose the exponent of e e as u u.
Integrate the expression in u and then substitute the original expression in x back into the u integral:
Let `u = -Tx` then `du = -T\ dx`. Our integral is now: `E=aint_0^(I_0)e^(-tx)dx` `=-a/Tint_(x=0)^(x=I_0)e^udu` `=-a/T[e^u]_(x=0)^(x=I_0)` `=-a/T[e^(-Tx)]_0^(I_0)` `=-a/T(e^(-TI_0)-e^0)` `=a/T(1-e^(-TI_0))`
Free Integral Calculator helps you solve definite and indefinite integration problems. Also double, triple and improper integrals. Answers, graphs, alternate forms.
The following problems involve the integration of exponential functions. We will assume knowledge of the following well-known differentiation formulas : where a is any positive constant not equal to 1 and is the natural (base e) logarithm of a .
