Integration by Algebraic Substitution.

#Integration #by #Substitution

Substitution for integrals corresponds to the chain rule for derivatives.

Suppose that F(u) is an antiderivative of f(u):

∫f(u)du=F(u)+C.

Assuming that u=u(x) is a differentiable function and using the chain rule, we have

ddxF(u(x))=F′(u(x))u′(x)=f(u(x))u′(x).

Integrating both sides gives

∫f(u(x))u′(x)dx=F(u(x))+C.

Hence

∫f(u(x))u′(x)dx=∫f(u)du,whereu=u(x).

This is the substitution rule formula for indefinite integrals.

Note that the integral on the left is expressed in terms of the variable x. The integral on the right is in terms of u.

The substitution method (also called u−substitution) is used when an integral contains some function and its derivative. In this case, we can set u equal to the function and rewrite the integral in terms of the new variable u. This makes the integral easier to solve.

Do not forget to express the final answer in terms of the original variable x.


If you find this video interesting, kindly subscribe to my channel for more exciting Maths tutorials.

[jcby]

THE MOST HELPFUL OUT OF ALL I'VE WATCHED, LOVE YAA@

[ugiribassey]

I always enjoy your lectures, tnx much.

[kensonmalupande]

This is excellent, sir👍👍💥 please keep it up

[rotimibukunmi]

Including the one you were doing is dat the final answer?

[rotimibukunmi]

Thank you sir

Pls you didn’t finish those questions

[elamahfalihah]

Why is it raise to the power of 3 instead of 3/2

[jinjenny]

Pls where is the half from in equation 1

[thadeemabirankunda]

Always we Can fixe u= x or we Can change? For exemple: U= cos²x.It' s possible?

[odogwunwoke]

hey good morning sir, sorry but why did you set u = x^2 in the first equation and not e^x

[elamahfalihah]

Why is it raise to the power of 3 instead of 3/2