# How do you integrate #7/(x^2+1)# using partial fractions?

This integral would normally be done using trigonometric substitution, but if you really want to use partial fractions to integrate this then you will need to use Complex coefficients:

Equating coefficients:

By signing up, you agree to our Terms of Service and Privacy Policy

To integrate ( \frac{7}{x^2+1} ) using partial fractions, follow these steps:

- Factor the denominator: ( x^2+1 ) cannot be factored further over the real numbers.
- Write the fraction in the form of partial fractions: ( \frac{7}{x^2+1} = \frac{A}{x} + \frac{B}{x^2+1} ).
- Clear the denominators by multiplying both sides by ( x^2 + 1 ).
- Equate coefficients to find the values of ( A ) and ( B ).
- Integrate each term separately.
- Combine the integrals and simplify if necessary.

Let's proceed with the steps:

- ( x^2+1 ) cannot be factored further over the real numbers.
- Write the fraction in the form of partial fractions: ( \frac{7}{x^2+1} = \frac{A}{x} + \frac{B}{x^2+1} ).
- Clear the denominators by multiplying both sides by ( x^2 + 1 ): [ 7 = A(x^2 + 1) + Bx ]
- Equate coefficients to find the values of ( A ) and ( B ):
- For ( x^2 ) term: ( A = 0 ).
- For ( x ) term: ( B = 7 ).

- Integrate each term separately:
- Integral of ( \frac{7}{x^2+1} ) becomes ( 7 \arctan(x) ).
- Integral of ( \frac{7}{x} ) becomes ( 7 \ln|x| ).

- Combine the integrals: [ \int \frac{7}{x^2+1} , dx = 7 \arctan(x) + 7 \ln|x| + C ]

where ( C ) is the constant of integration.

By signing up, you agree to our Terms of Service and Privacy Policy

- How do you use sigma notation to write the sum for #3-9+27-81+243-729#?
- How do you evaluate the definite integral by the limit definition given #int x^2+1dx# from [1,2]?
- How do you integrate #int sqrttanxsec^2xdx#?
- How do you find the derivative of #F(x) = int sqrt(1+sec(3t)) dt#?
- How do you find the partial sum of #Sigma (4.5+0.025j)# from j=1 to 200?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7