Here are the direct answers to the questions in the provided image:

1. $\int \frac{x^4}{x^2 + 1} dx = \frac{x^3}{3} - x + \ln(x^2 + 1) + C$
2. $-\int \sin 3x \sin^2 x \, dx = \frac{-1}{12} \cos 3x + \frac{1}{8} \cos 5x - \frac{1}{24} \cos x + C$
3. $\int \frac{dx}{1 + \cos^2 x} = \frac{1}{\sqrt{2}} \tan^{-1} \left( \frac{\tan x}{\sqrt{2}} \right) + C$
4. $\int \sin x \sin nx \, dx = \frac{\sin(n-1)x}{2(n-1)} - \frac{\sin(n+1)x}{2(n+1)} + C$
5. If $\cosh x = \frac{4}{5}$, then $\cosh 2x = \frac{41}{25}$
6. $\frac{d}{dx} \int_0^x \sin^3 t \, dt = \sin^3 x$
7. $\int \sqrt{x^2 - 4} \, dx = \frac{x}{2} \sqrt{x^2 - 4} - 2 \ln \left( x + \sqrt{x^2 - 4} \right) + C$
8. $\int_0^{\pi/2} \frac{x}{\sin x + \cos x} \, dx = \frac{\pi}{4} \ln 2$
9. $-\int \sin^n x \, dx = \frac{-1}{n} \sin^{n-1} x \cos x + \frac{n-1}{n} \int \sin^{n-2} x \, dx$
10. $\int \sin^4 x \cos^5 x \, dx = \frac{\cos^6 x}{6} - \frac{\cos^8 x}{8} + \frac{\cos^{10} x}{10} + C$
11. $\int \sin^{10} x \, dx = \frac{1}{2} \int (1 - \cos 2x)^5 \, dx$
12. The inverse of the function $f(x) = x^2 - 3x + 2$ is $f^{-1}(x) = \frac{3 \pm \sqrt{9 - 4(x-2)}}{2}$
13. The inverse of $f(x) = \ln(x + 1)$ is $f^{-1}(x) = e^x - 1$
14. $\int (\cosh x + \sinh x)^4 \, dx = \int e^{4x} \, dx = \frac{e^{4x}}{4} + C$
15. $\int \sqrt{x^2 + 8x + 7} \, dx = \frac{(x+4)\sqrt{x^2 + 8x + 7}}{2} - \ln\left| x + 4 + \sqrt{x^2 + 8x + 7} \right| + C$
16. The range of the function $f(x) = \sqrt{x^2 - 3x + 2}$ is $[0, +\infty)$
17. $\int \frac{x}{x^3 + 3} \, dx = \frac{1}{3} \ln |x^3 + 3| + C$
18. $\int \frac{3x + 5}{x^2 - 5x + 4} \, dx = \ln|x-4| - \ln|x-1| + C$
19. Approximating lines of the function $f(x) = \frac{8}{x^2 - 1}$: $\text{Asymptotes are } y = 0 \text{ and } x = \pm 1$

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Related Questions:

1. How is the range of a square root function determined?
2. What are common techniques to compute definite integrals with limits?
3. How is the inverse of a logarithmic function derived?
4. Why are trigonometric identities essential in integral computation?
5. How do you solve integrals involving hyperbolic functions?

Tip:

When solving definite integrals, always check the boundaries to ensure that the function is defined and continuous within the interval.