That looks good so far. What about splitting up the integral into two now?Math9999 said:Homework Statement
Find the integral of 1/(1+cosx) dx from -pi/2 to pi/2.
Homework Equations
None.
The Attempt at a Solution
Here's my work:
1/(1+cosx)=(1-cosx)/((1+cosx)(1-cosx))=(1-cosx)/(1-cos^2 x)=(1-cosx)/sin^2 x
This is what I've got so far. But this doesn't seem to simplify the integrand.
There was a quicker way. Hint: ##x = 2(x/2)##.Math9999 said:Never mind. I got it. You gave me the big hint already, splitting it up.
To find the integral of a basic function, you can use the power rule, which states that the integral of x^n is (x^(n+1))/(n+1). For example, the integral of x^2 would be (x^3)/3. You can also use the table of basic integrals to find the integral of common functions such as sine, cosine, and logarithms.
To find the integral of a trigonometric function, you can use trigonometric identities and substitution. For example, if the integral is of the form ∫sin(x)dx, you can use the identity sin^2(x) + cos^2(x) = 1 to rewrite it as ∫sin^2(x)dx + ∫cos^2(x)dx. Then, you can use the substitution u = sin(x) or u = cos(x) to solve each integral separately.
The process for finding the integral of a rational function involves using partial fractions and integration by parts. First, you need to factor the denominator of the rational function and use the partial fractions method to break it down into simpler fractions. Then, you can use integration by parts to find the integral of each fraction. Finally, you can combine the results to find the integral of the original rational function.
To find the definite integral of a function, you need to evaluate the indefinite integral at the upper and lower limits of integration and then subtract the lower limit from the upper limit. This will give you the area under the curve between the two limits. You can also use the fundamental theorem of calculus, which states that the definite integral of a function is equal to the difference between the antiderivative at the upper limit and the antiderivative at the lower limit.
Some common techniques for finding integrals include substitution, integration by parts, partial fractions, and trigonometric identities. Other techniques include using symmetry, trigonometric substitutions, and the method of undetermined coefficients. It is also helpful to have a good understanding of basic integrals and their properties, as well as knowing when to use different techniques for different types of integrals.