Integral of e sin x is a function that involves the exponential function (e), the sine function (sin), and the integration operation (integral). It represents the area under the curve of e sin x over a specified interval. Evaluating this integral requires techniques such as integration by parts. This integral is also related to the complex exponential function, which is a combination of e and sin x.
How to Integrate e^sin x
Integrals of the form $$\int e^{\sin x} dx$$ can be tricky to solve, but there are a few different techniques that can be used to find the answer.
1. Substitution
One way to solve this integral is to use substitution. Let $u = \sin x$. Then $du = \cos x dx$, and the integral becomes
$$\int e^{\sin x} dx = \int e^u du = e^u + C = e^{\sin x} + C$$
2. Integration by parts
Another way to solve this integral is to use integration by parts. Let $u = e^{\sin x}$ and $dv = dx$. Then $du = e^{\sin x} \cos x dx$ and $v = x$. Using the formula for integration by parts, we have
$$\int e^{\sin x} dx = uv – \int v du = xe^{\sin x} – \int x e^{\sin x} \cos x dx$$
Now, we can use substitution again to solve the integral on the right-hand side. Let $w = \sin x$. Then $dw = \cos x dx$, and the integral becomes
$$\int x e^{\sin x} \cos x dx = \int x e^w dw = xe^w – \int e^w dx = xe^{\sin x} – e^{\sin x} + C$$
Substituting this back into the original equation, we have
$$\int e^{\sin x} dx = xe^{\sin x} – \int x e^{\sin x} \cos x dx = xe^{\sin x} – xe^{\sin x} + e^{\sin x} + C = e^{\sin x} + C$$
3. Trigonometric identities
A third way to solve this integral is to use trigonometric identities. We can use the identity $\sin^2 x + \cos^2 x = 1$ to rewrite the integral as follows:
$$\int e^{\sin x} dx = \int e^{\sin x} \frac{\cos^2 x + \sin^2 x}{\cos^2 x} dx = \int e^{\sin x} \left(1 + \tan^2 x\right) dx$$
Now, we can use substitution again to solve the integral on the right-hand side. Let $z = \tan x$. Then $dz = \sec^2 x dx$, and the integral becomes
$$\int e^{\sin x} \left(1 + \tan^2 x\right) dx = \int e^{\sin x} \left(1 + z^2\right) dz = e^{\sin x} z – \int e^{\sin x} z dz$$
Now, we can use integration by parts again to solve the integral on the right-hand side. Let $u = z$ and $dv = e^{\sin x} dz$. Then $du = dz$ and $v = e^{\sin x}$. Using the formula for integration by parts, we have
$$\int e^{\sin x} z dz = ze^{\sin x} – \int e^{\sin x} dz = ze^{\sin x} – e^{\sin x} + C$$
Substituting this back into the original equation, we have
$$\int e^{\sin x} dx = \int e^{\sin x} \left(1 + \tan^2 x\right) dx = e^{\sin x} \tan x – \int e^{\sin x} \tan x dx = e^{\sin x} \tan x – e^{\sin x} + C$$
Question 1:
What is the general formula for integrating e sin x?
Answer:
The general formula for integrating e sin x is:
∫ e sin x dx = (1/2)e sin x – (1/2)e cos x + C
where C is an arbitrary constant.
Question 2:
Explain the steps involved in integrating e sin x.
Answer:
The steps involved in integrating e sin x are:
- Use the u-substitution method, where u = sin x.
- This results in du/dx = cos x and dx = du/cos x.
- Substitute u and du/cos x into the integral: ∫ e sin x dx = ∫ e^u du.
- Integrate e^u with respect to u, which results in e^u + C.
- Substitute back u = sin x and simplify to get the final answer: (1/2)e sin x – (1/2)e cos x + C.
Question 3:
What are the applications of integrating e sin x?
Answer:
The applications of integrating e sin x include:
- Solving differential equations.
- Modeling sinusoidal oscillations.
- Calculating the Fourier transform of a signal.
- Analyzing electrical circuits.
And there you have it, folks! That’s all you need to know about integrating e^sin(x). It’s not the easiest integral, but with a little patience and the right techniques, you can tame this mathematical beast. Thanks for reading, and keep practicing. The more you do, the easier it will become. See you next time for another exciting journey into the world of calculus!