The non definite integral of (e ^ x) * (sinx) ^ 2
时间: 2024-05-26 22:15:33 浏览: 21
Let's start by using integration by parts:
Let u = sin^2(x) and dv/dx = e^x
Then du/dx = 2sin(x)cos(x) and v = e^x
Using the integration by parts formula, we have:
∫ (e^x)*(sin^2(x)) dx = e^x*sin^2(x) - ∫ 2e^x*sin(x)cos(x) dx
Now, we can use substitution to simplify the integral on the right-hand side. Let u = sin(x) and du/dx = cos(x)dx. Then we have:
∫ 2e^x*sin(x)cos(x) dx = 2∫ e^x*sin(x)*cos(x)*(cos(x)/cos(x)) dx
= 2∫ e^x*sin(x)*cos^2(x)/cos(x) dx
= 2∫ e^x*sin(x)*(1-sin^2(x)) dx/cos(x)
= -2∫ e^x*sin^3(x) dx/cos(x)
Now, we can use integration by parts again. Let u = sin^3(x) and dv/dx = e^x/cos(x). Then du/dx = 3sin^2(x)cos(x) and v = ln|sec(x) + tan(x)|. We have:
-2∫ e^x*sin^3(x) dx/cos(x) = -2ln|sec(x) + tan(x)|*e^x*sin^3(x) + 6∫ ln|sec(x) + tan(x)|*e^x*sin^2(x)cos(x) dx
This integral does not have a closed-form solution in terms of elementary functions. Therefore, this is the final answer:
∫ (e^x)*(sin^2(x)) dx = e^x*sin^2(x) + 2ln|sec(x) + tan(x)|*e^x*sin^3(x) - 6∫ ln|sec(x) + tan(x)|*e^x*sin^2(x)cos(x) dx
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