Integration by substitution method

On the basis of the method is following simple feature of indefinite integral:

variable substitution in the integral

We express initial integration variable x in terms of new variable t and get the expression for dx. Then we substitute derived expression into initial integral. We assume, that new variable is fitted such as new integral has the simplier form than initial.

Example:

sin3xcosxdxtsinxdtcosxdxt3dtt44Constsin4x4Const

we introduce new variable by using the formula:

tsinx

then calculate the expression dt:

differential of the new variable

after this, we substitute new expression into initial integral:

expression of the integral in terms of new variable

When the change of variable is done, we get more simple integral:

t3dt

which easily can be integrated:

t3dtt44Const

after calculations is done, we should go back to the old variable:

t44tsinxsin4x4

thus, finally we get:

sin3xcosxdxsin4x4Const

Derived result can always be checked by the differentiation:

verification of the integral solution

Next example:

1x13xdxxt6t6xdx6t5dt6t5t613t6dt6t5t31t2dt6t21t2dt

continue:

6111t2dt61dt11t2dt6tarctgtConst66xarctg6xConst

Use our online integrals calculator which automatically determines and makes optimal variable substitution to calculate your integral with step by step solution.

Other useful links:

Derivatives table
Area between crossed curves online calculator

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