7. Antiderivatives & Indefinite Integrals

Evaluate the indefinite integral as a power series: $\int \arctan \left(x\right)x dx$

Find $h\(\left\)(x\(\right\))$ by evaluating the following indefinite integral.

$h\left(x\right)=25x^{4}dx$

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ sec² s/10 ds

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ ((e²ʷ - 5eʷ + 4)/(eʷ - 1))dw

Using different substitutions

Show that the integral

∫((x² - 1)(x + 1))^(-2/3) dx

can be evaluated with any of the following substitutions.

a. u = 1/(x + 1)

What is the value of the integral?

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ √(x - x²) / x dx

What change of variables would you use for the integral ∫(4 - 7x)^(-6) dx?

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ ((4x⁴ - 6x²) / x ) dx

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ x (7x + 5)^(3/2) dx

1. Give some examples of analytical methods for evaluating integrals.

Evaluate the integral: ${\int }_0^{\frac{\pi }{2}}9{\cos }^2\left(x\right) dx$.

Evaluate the indefinite integral: ${\int }_{}^{}5x^2+25 dx$

Evaluate the indefinite integral: $\int (9x^2+81)dx$

71. Different Methods

Let I = ∫ (x²)/(x + 1) dx.

b. Evaluate I by first performing long division on the integrand.

Evaluate the indefinite integral. Remember to include the constant of integration.
$${\int }_{}^{}{(\ln (x\left)\right)}^{2}dx$$