- #1
Memo
- 35
- 3
- Homework Statement
- ∫(sinx+sin^3x)dx/(cos2x)
- Relevant Equations
- cos2x=2cos^2x-1
Could you check if my answer is correct? Thank you very much!
Is therea simpler way to solve the math?
You can do that yourself by differentiating your answer and checking you get the original integrand.Memo said:Homework Statement: ∫(sinx+sin^3x)dx/(cos2x)
Relevant Equations: cos2x=2cos^2x-1
View attachment 334635
Could you check if my answer is correct?
Your method looks good to me. Maybe there's a trick, but not always.Memo said:Thank you very much!
Is therea simpler way to solve the math?
Could you tell me how?PeroK said:You can do that yourself by differentiating your answer and checking you get the original integrand.
PeroK said:You can do that yourself by differentiating your answer and checking you get the original integrand.
If you integrate a function f(x) and get an antiderivative F(x) + C, you can check your answer by differentiating F(x). If your antiderivative is correct, the result will be f(x).Memo said:Could you tell me how?
It appears that I was wrongMark44 said:If you integrate a function f(x) and get an antiderivative F(x) + C, you can check your answer by differentiating F(x). If your antiderivative is correct, the result will be f(x).
In symbols...
If ##\int f(x) dx = F(x) + C##, then ##\frac d{dx}\left(F(x) + C\right) = f(x)##
Look up the Fundamental Theorem of Calculus.Mark44 said:If you integrate a function f(x) and get an antiderivative F(x) + C, you can check your answer by differentiating F(x). If your antiderivative is correct, the result will be f(x).
In symbols...
If ##\int f(x) dx = F(x) + C##, then ##\frac d{dx}\left(F(x) + C\right) = f(x)##
Memo said:Homework Statement: ∫(sinx+sin^3x)dx/(cos2x)
Relevant Equations: cos2x=2cos^2x-1
View attachment 334635
Could you check if my answer is correct? Thank you very much!
Is therea simpler way to solve the math?
An integral involving powers of trig functions is an integral that includes trigonometric functions such as sine, cosine, tangent, etc., raised to a power. These types of integrals often require the use of trigonometric identities and substitution techniques to evaluate.
To solve integrals involving powers of trig functions, you typically use trigonometric identities to simplify the expression and then apply integration techniques such as substitution or integration by parts. It may also be helpful to rewrite the trigonometric functions in terms of exponentials using Euler's formula.
Some common trigonometric identities used in integrals involving powers of trig functions include Pythagorean identities (sin^2(x) + cos^2(x) = 1), double angle identities, half angle identities, and power-reducing identities. These identities help simplify trigonometric expressions and make them easier to integrate.
Yes, there are special techniques for evaluating integrals involving powers of trig functions. One common technique is to use trigonometric substitution, where you substitute trigonometric functions to simplify the integral. Another technique is to use partial fractions or integration by parts, depending on the complexity of the trigonometric expression.
Integrals involving powers of trig functions are important in mathematics because they arise in various applications such as physics, engineering, and signal processing. These integrals help in solving problems related to periodic functions, vibrations, and waveforms, making them essential in many areas of science and technology.