u-Substitution

Front

What is u-substitution in integration?

Back

U-substitution is a method used to simplify the process of integration by substituting a part of the integrand with a new variable (u), making the integral easier to solve.

Front

How do you choose u in u-substitution?

Back

Choose u to be a function inside the integral that simplifies the expression when differentiated. Common choices include polynomials or expressions that are raised to a power.

Front

What is the derivative of u with respect to x in u-substitution?

Back

If u = g(x), then the derivative is du/dx = g'(x). This relationship is used to replace dx in the integral.

Front

What is the formula for u-substitution?

Back

If u = g(x), then the integral can be transformed as follows: \int f(g(x))g'(x)dx = \int f(u)du.

Front

Example of u-substitution: Evaluate \int 6x(x^2 + 1)^2 dx.

Back

Let u = x^2 + 1, then du = 2x dx. The integral becomes 3\int u^2 du = (x^2 + 1)^3 + C.

Front

What is the integral of a constant multiplied by a function?

Back

The integral of a constant a multiplied by a function f(x) is given by \int a f(x) dx = a \int f(x) dx.

Front

How do you handle definite integrals with u-substitution?

Back

When using u-substitution for definite integrals, change the limits of integration to correspond to the new variable u.

Front

What is the integral of \int 6x(3x^2 + 4)^4 dx?

Back

Using u-substitution with u = 3x^2 + 4, the integral evaluates to \frac{6(3x^2 + 4)^5}{5} + C.

Front

What is the integral of \int 3x^2 \sqrt{x^3 - 2} dx?

Back

Using u-substitution with u = x^3 - 2, the integral evaluates to \frac{2}{3}(x^3 - 2)^{\frac{3}{2}} + C.

Front

What is the importance of the constant of integration C?

Back

The constant of integration C represents the family of antiderivatives and accounts for any constant that may have been lost during differentiation.