Before you go straight to the complex ways of solving integrals try these simple algebra concepts.
Expand the Integral: |
Separate the Fractions: |
Remember when solving integrals that if you have an equation that is a quantity to an exponent, you can foil the equation and then take the integral once expanded.
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If you have a fraction with a sum in the numerator you can break the fraction into a sum of multiple fractions.
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Divide: |
Conjugate the Denominator: |
If you have an improper fraction you can divide the fraction to get a new equation with a remainder.
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Another thing you can try to do in order to solve an integral is conjugating the denominator. To do so multiply the denominator by it's opposite over it's opposite. (If you have 1/(x+y) multiply by (x-y)/(x-y))
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Substitution:
To solve by substitution
1.) Pick a part of the equation to substitute for a variable. Set that part of the equation equal to a variable. (For example purposes we will use the letter u.) 2.) Solve for the derivative of the equation that you have created 3.) Manipulate the equation so that you can substitute du into the equation. Multiply the inside by the number needed inside the integral and divide by the same number on the outside of the integral. 4.) Substitute u and du into the integral 5.) Solve the integral with the new variables. Don't forget +C! 6.) Substitute the equation back in for the variable |
= -ln|4-x| + C |
1. Pick the 4-x and set it equal to u
2. Take the derivative of u = 4-x 3. Multiply the inside by -1 and divide the outside by -1 4. Substitute 5. Solve 6. Substitute 4-x back in |
*If you are solving for a definite integral, plug the limits into the equation for u to find new limits for dealing with u. After taking the integral, don't have a +C. Plug the upper limit into the equation and subtract the equation with the lower limit in it.
Change of Variables:
To solve by change of variable
1.) Pick a part of the equation to substitute for a variable. Set that part of the equation equal to a variable. (For example purposes we will use the letter u.) 2.) Manipulate the equation so that it is x in terms of u 3.) Solve for the derivative of the equation that you have created 4.) Substitute u, x and dx into the integral 5.) Simplify the equation 6.) Solve the integral with the new variables. Don't forget +C! 7.) Simplify the equation 8.) Substitute the equation back in for the variable |
1. Pick the square root of x-1 to set equal to u
2. Square both sides and then add 1 to both sides to get the equation of x in terms of u 3. Take the derivative of the x equation 4. Substitute u in for the square root of x-1, u squared +1 in for x, and 2udu in for dx 5. Bring the 2 to the outside of the integral and simplify the equation 6. Solve 7. Simplify 8. Substitute the square root of x-1 back into the equation |
*If you are solving for a definite integral, plug the limits into the equation for u to find new limits for dealing with u. After taking the integral, don't have a +C. Plug the upper limit into the equation and subtract the equation with the lower limit in it.
Partial Fractions:
1.) Simplify the denominator of the equation as much as you can.
2.) For each of the factors in the denominator create a new fraction with a variable in the top. 3.) Make an equation where you have the original fraction equal to the sum of the new fractions 4.) Multiply both sides by the original denominator. 5.) Substitute a number in for x that will make a factor equal to 0. 6.) Solve for a variable 7.) Repeat steps 5 & 6 with different variables until you have values for all variables 8.) Take the integral of the side of equation from step 3 without the original equation. Plug in the values for the variables. 9.) Solve the integral |
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1. Simplify the denominator
2. Set up the equation 3. Multiply by the denominator 4. Substitute -2 in for x 5. Solve to get B = -1 4. Substitute 3 in for x 5. Solve to get A = 2 6. Plug 2 and -1 in for A and B from the third line 7. Take the integral |
Integration by Parts:
1.) Pick part of the integral to set equal to u, and set the other part of the equation (including the dx) equal to dv.
*In most cases use the more complicated part that can be integrated easily for dv and the part that simplifies when differentiating for u ***note sometimes the parts of the equations that you choose for u and dv don't work and you have to switch which ones you use for which 2.) Take the derivative of the u equation and the integral of the dv equation. 3.) Multiply u by v and subtract the integral of v times dv 4.) Simplify repeating as necessary |
1. Pick x squared to be equal to u and cos(x) to be equal to dv
2. Take the derivative of x squared and the integral of cos(x) 3. Plug u, v, and dv into uv-∫vdv 4. To simplify the equation you need to reuse integration by parts for the integral. 1. Set 2x equal to u and sin(x) equal to dv 2. Take the derivative of 2x and the integral of sin(x) 3. Plug u, v, and dv into uv-∫vdv 4. Simplify |