7. Antiderivatives & Indefinite Integrals

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(sin2x − csc²x)dx

Evaluate the integral. (Use c for the constant of integration.) $\int \ln \left(x\right) dx$

Evaluate the indefinite integral: ${\int }_{}^{}\frac{5}{1+x^2}dx$

Evaluate the indefinite integral as an infinite series: ${\int }_{}^{}\frac{(cosx-1)}{x}dx$.

Find the general indefinite integral. (Use c for the constant of integration.) $4x^5 \int dx$

7–64. Integration review Evaluate the following integrals.

20. ∫ eˣ (1 + eˣ)⁹ (1 - eˣ) dx

Evaluate the integral. (Use $c$ for the constant of integration.) $\int x{x}^4+16dx$

Using an upper-case C for any arbitrary constants, find the general indefinite integral: ${\int }_{}^{}{x}^{2}dx$ =

Evaluate the indefinite integral: $0.75x^2-16x^2 dx$

Evaluate the integral. (Use c for the constant of integration.) ${\int }_{}^{}{(\ln x)}^{2}dx$

Evaluate the integral. (Use c for the constant of integration.) $\int 9{\sin }^2\left(x\right){\cos }^3\left(x\right) dx$

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ (x² dx) / (x² - 1)^(5/2), where x > 1

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

24. ∫ dt / √(1 + 4eᵗ)

Evaluate the indefinite integral: $\int (2x^2+3x+1)dx$

General results Evaluate the following integrals in which the function ƒ is unspecified. Note that ƒ⁽ᵖ⁾ is the pth derivative of ƒ and ƒᵖ is the pth power of ƒ. Assume ƒ and its derivatives are continuous for all real numbers. 

∫ (5 ƒ³ (𝓍) + 7ƒ² (𝓍) + ƒ (𝓍 )) ƒ'(𝓍) d𝓍