Integration by Substitution - choosing a u

Identify the u and the du:

• u=10x, du = 10dx

• u = 5x² + 1, du = 10xdx

• u = (5x² +1)², du = 10xdx

• u = (5x² +1)², du = 2(5x² + 1) 10xdx

Answer explanation

The correct choice is u = 5x² + 1, du = 10xdx. When you substitute, it will become u to the 2nd power, and the x with cancel with the du.

• u = x²

• u = x

• u = x - 4

Answer explanation

The correct choice is u= x - 4. When you make the substitution, the x² does not cancel, so you will need to solve for x in terms of u, then simplify.

Identify the u and the du.

• u = -x², du = -2x dx

• u = 1 - x², du = -2xdx

• u = -2x, du = -2dx

Answer explanation

The correct choice is u = 1 - x², du = -2xdx. This substitution simplifies integration, as it directly relates to the derivative of u with respect to x, making it easier to work with in calculus problems.

Identify the u and the du.

• u = 3x², du = -6x dx

• u = x³ + 1, du = 3x² dx

Answer explanation

The correct choice is u = x^3 + 1, du = 3x^2 dx. Here, u is a simple polynomial, and its derivative, du, is calculated using the power rule, confirming the relationship between u and du.

Identify the u and the du.

Answer explanation

The correct choice is u=(4+1/x^2) and du=-2/x^3 dx. This is derived from differentiating u with respect to x, applying the chain rule.

Identify the u and the du.

• u = 1+2x, du = 2dx

• u = 2, du = 0dx

• u = 2x, du = 2dx

Answer explanation

The correct choice is u = 1+2x, du = 2dx. Here, u is defined as a function of x, and du represents its derivative with respect to x, which is calculated as 2dx.

• u = x

Answer explanation

The correct choice is u = 2x^2, so that the du = 4x will cancel the x on the top. You will then create inverse sine as the integral to solve.

Identify the u and the du.

Answer explanation

The correct choice is u=1+√x because it simplifies the integration process. The derivative du=1/(2√x)dx is derived from differentiating u with respect to x, making it the appropriate substitution.

Identify the u, and the du.

Answer explanation

The correct choice is u=4-\sqrt{x}, du=-\frac{1}{2\sqrt{x}}dx. This is derived by differentiating u with respect to x, which gives the correct expression for du.

Identify the u and the du.

• u = x, du = dx

• u = x -1, du = dx

• u = (x - 1)⁴, du = dx

• u = (x - 1)⁴, du = 4(x - 1)³dx

Answer explanation

The correct choice is u = x - 1, du = dx because the derivative of u with respect to x is simply 1, making du equal to dx. Other options either misrepresent u or incorrectly calculate du.