# How do you integrate #int5xsqrt(2x+3)# using substitution?

This integral doesn't seem quiet obvious, so I will go through all the steps in high detail.

When evaluating integrals, it is sometimes a matter of trial and error, especially when it comes to u-substitutions.

Proceeding with the integration process, we can now rewrite our integral.

If we multiply everything into the parentheses and simplify we get

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

To integrate the function ( \int 5x \sqrt{2x + 3} ) using substitution, we can let ( u = 2x + 3 ). Then, ( du/dx = 2 ), which implies ( dx = du/2 ). Substituting these values into the integral gives us:

[ \int 5x \sqrt{2x + 3} , dx = \int 5x \sqrt{u} , \frac{du}{2} ]

Now, we can simplify and solve this new integral:

[ \frac{5}{2} \int x \sqrt{u} , du ]

To integrate ( x \sqrt{u} ), we use integration by parts with ( u = x ) and ( dv = \sqrt{u} , du ). This gives us ( du = dx ) and ( v = \frac{2}{3} u^{3/2} ).

Applying integration by parts:

[ \frac{5}{2} \left( x \cdot \frac{2}{3} u^{3/2} - \int \frac{2}{3} u^{3/2} , dx \right) ]

Simplifying further:

[ \frac{5}{3} xu^{3/2} - \frac{10}{9} \int u^{3/2} , du ]

Integrating ( u^{3/2} ) gives us ( \frac{2}{5} u^{5/2} ). Substituting back ( u = 2x + 3 ) gives us:

[ \frac{5}{3} xu^{3/2} - \frac{20}{45} (2x + 3)^{5/2} + C ]

Where ( C ) is the constant of integration. Finally, simplifying the expression:

[ \frac{5}{3} x(2x + 3)^{3/2} - \frac{40}{45} (2x + 3)^{5/2} + C ]

This is the result of integrating ( \int 5x \sqrt{2x + 3} ) using substitution and integration by parts.

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

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