8. Definite Integrals

Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.

∫ (from 0 to 1/√3) dt / (t² + 1)^(7/2)

7–84. Evaluate the following integrals.

11. ∫ from 0 to π/4 (sec x – cos x)² dx

Evaluate the indefinite integral.

${\int }_{}^{}\frac{1}{{(3x+2)}^5}dx$

Evaluate the indefinite integral.

${\int }_{}^{}\theta \bullet se{c}^2(5{\theta }^2+1)d\theta$

Evaluate the indefinite integral.

${\int }_{}^{}x{(5+x)}^{79}dx$

Evaluate the indefinite integral.

${\int }_{}^{}\frac{t}{\sqrt{t-2}}dt$

Evaluate the indefinite integral.

${\int }_{}^{}\frac{x}{{(x-6)}^5}dx$

Evaluate the indefinite integral.

${\int }^{}sin\theta {(1+\sin \theta )}^{84}\cos \theta d\theta$

Evaluate the definite integral.

${\int }_0^1\frac{t}{\sqrt{t^2+1}}dt$

Evaluate the definite integral.

${\int }_1^2(x-3){(x^2-6x)}^{2}dx$

Evaluate the definite integral.

${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\tan y$ $se{c}^{2}ydy$

Evaluate the indefinite integral.

${\int }_{}^{}3t\sqrt{t^2+7}dt$

7–64. Integration review Evaluate the following integrals.

16. ∫ from 0 to 1 of (t² / (1 + t⁶)) dt

7–64. Integration review Evaluate the following integrals.

18. ∫ from 3 to 7 of (t - 6) * √(t - 3) dt

7–64. Integration review Evaluate the following integrals.

32. ∫ from 0 to 2 of x / (x² + 4x + 8) dx