Integration by Substitution


Integration by Substitution/Replacement
The integration by substitution rule applies if the integrand consists two parts (functions) in which one function is derivative of other (in some form). The two general structure of this form is
f(x)n.f(x)dx or f(x)dxf(x)n
That is, if a one part is the derivative of another via the chain rule, we let u to denote a likely candidate for the inner part as f(x), then translate the given function so that it is written entirely in terms of u, with no x remaining in the expression.
If we can integrate this new function of u, then the anti-derivative of the original function is obtained by replacing u by the equivalent expression in x.
It can be further classified into three sub categories
  1. General substitution
  2. Powers of sine and cosine
  3. Trigonometric Substitutions



  1. dxQuadratic
  2. linearQuadraticdx
  3. dxQuadraticdx
  4. linearQuadraticdx
  1. dxx2a2dx=12alog|xax+a|+c
  2. dxa2x2dx=12alog|a+xax|+c
  3. dxx2+a2dx=1atan1(xa)+c
  4. dxa2x2dx=sin1(xa)+c
  5. dxx2a2dx=log|x+x2a2|+c
  6. dxx2+a2dx=log|x+x2+a2|+c
  7. dxxx2a2dx=1asec1(xa)+c



Solved Examples: General substitution

Evaluate Following Integrals
  1. (x25)72xdx

    Solution 👉 Click Here

  2. 2x(cosx2)dx

    Solution 👉 Click Here

  3. (ax+b)ndx, assuming that a and b are constants, a0, and n is a positive integer.

    Solution 👉 Click Here

  4. sin(ax+b)dx, assuming that a and b are constants and a0.

    Solution 👉 Click Here




Evaluate the following Intrgral

SN Question Answer
1 sin2xdx x2sin(2x)4+c
2 sin3xdx cosx+cos3x3+c
3 sin4xdx 3x8sin(2x)4+sin(4x)32+c
4 cos2xsin3xdx cos5x5cos3x3+c
5 cos3xdx sinxsin3x3+c
6 cos3xsin2xdx sin3x3sin5x5+c



Solved Examples: Powers of sine and cosine

Evaluate the following integrals
  1. sin5xdx.

    Solution 👉 Click Here

  2. sin2xcos2xdx.

    Solution 👉 Click Here

  3. tan4xsec2xxdx

    Solution 👉 Click Here

  4. csc6dx

    Solution 👉 Click Here

  5. sin(x+a)sin(xa)dx

    Solution 👉 Click Here

Trigonometric Substitutions

This type of substitution is usually indicated when the function of integrate contains a polynomial expression to use the fundamental trigonometric identity as
cos2x+sin2x=1 or cos2x=1sin2x
sec2xtan2x=1 or sec2x=1+tan2x
sec2xtan2x=1 or tan2x=sec2x1

Therefore
  1. if it contains 1x2, we substitute x=sinu
  2. if it contains 1+x2 we substitute x=tanu and
  3. if it contains x21, we substitute x=secu.

Sometimes, we need something a bit different to handle constants other than one. Thus, key points are as follows

α
x
1-x2
1
α
x
1
1+x2
α
x^2-1
1
x
  1. Put x=asinu for integration involving a2x2
  2. Put x=atanu for integration involving a2+x2
  3. Put x=asecu for integration involving x2a2

Solved Examples: Trigonometric Substitutions

Evaluate following integrals
  1. 1x2dx

    Solution 👉 Click Here

  2. 49x2dx

    Solution 👉 Click Here

  3. x24x2dx

    Solution 👉 Click Here

  4. xa3x3dx

    Solution 👉 Click Here




Evaluate the following Intrgral

SN Question Answer
1 1x2a2dx x9+x2dx
2 x21dx xx212ln|x+x21|2+c
3 9+4x2dx x9+4x22+94ln|2x+9+4x2|+c
4 x1x2dx (1x2)2/33+c
5 x21x2dx sin18sin(4sin11x)32+c
6 11+x2dx ln|x+1+x2|+c
7 x24x2dx 2sin1(x/2)x4x22+c
8 x34x21dx (2x2+1)4x2124+c

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