Example 1.20. Find $\int x^{1 / 4}\left(1+x^{1 / 2}\right)^{-7 / 2} d x$.
Solution. In the given case, $m=\frac{1}{4}, n=\frac{1}{2}$, $p=-\frac{7}{2}, \frac{m+1}{n}+p=-1-$ is an integer. Let $t=\sqrt{1+\frac{1}{x^{1 / 2}}}$.

Then $t^{2}=1+\frac{1}{x^{1 / 2}}, \quad x^{1 / 2}=\frac{1}{t^{2}-1}, \quad x=\frac{1}{\left(t^{2}-1\right)^{2}}, \quad d x=$ $=-4 t\left(t^{2}-1\right)^{-3} d t, \quad x^{1 / 4}=\frac{1}{\left(t^{2}-1\right)^{1 / 2}}, \quad 1+x^{1 / 2}=1+$ $+\frac{1}{t^{2}-1}=\frac{t^{2}}{t^{2}+1}$.

As a result, we get

$$
\begin{aligned}
& \int x^{1 / 4}\left(1+x^{1 / 2}\right)^{-7 / 2} d x= \\
& =\int \frac{1}{\left(t^{2}-1\right)^{1 / 2}}\left(\frac{t^{2}}{t^{2}-1}\right)^{-7 / 2}\left(-4 t\left(t^{2}-1\right)^{-3}\right) d t= \\
& \quad=-4 \int t^{-6} d t=\frac{4}{5 t^{5}}+C=\frac{4}{5} \frac{\sqrt[4]{5}}{\sqrt{\left(1+x^{1 / 2}\right)^{5}}}+C
\end{aligned}
$$

## 1.4.  PROBLEMS FOR CHAPTER 1

Find the integrals.
1.21. $\int\left(\frac{1-x}{x}\right)^{3} d t$
1.22. $\int 2^{x} 7^{x} d x$
1.23. $\int \frac{d x}{\sqrt{5-5 x^{2}}}$
1.24. $\int(\sqrt[3]{x}+1)\left(\sqrt[3]{x^{2}}-\sqrt[3]{x}+1\right) d x$
1.25. $\int \operatorname{tg}^{2} x, d x$
1.26. $\int 2 \cos ^{2} \frac{x}{2} d x$
1.27. $\int \frac{1=3 x^{2}}{x^{2}\left(1+x^{2}\right)} d x$
1.28. $\int(\arccos 3 x+\arcsin 3 x) d x$
1.29. $\int \cos x d(\cos x)$
1.30. $\int \operatorname{ctg}^{4} x d(\operatorname{ctg} x)$
1.31. $\int(x+10)^{1} 7 d x$
1.32. $\int \sqrt[3]{7-3 x} d x$
1.33. $\int \frac{x^{3}}{\sqrt{x^{4}+1}} d x$
1.34. $\int \frac{\sqrt[3]{\ln x}}{x} d x$
1.35. $\int \frac{\operatorname{arctg}^{5} x d x}{1=x^{2}}$
1.36. $\int \cos (3 x+4) d x$
1.37. $\int \sin (2-5 x) d x$
1.38. $\int \operatorname{tg} x d x$
1.39. $\int \operatorname{ctg} x d x$
1.40. $\int \operatorname{ctg}(3 x+2) d x$
1.41. $\int e^{\sin ^{2} x} \sin 2 x d x$
1.42. $\int \frac{d x}{\sqrt{e^{-x}-1}}$
1.43. $\int e^{x^{4}} x^{3} d x$
1.44. $\int \frac{3^{x} d x}{\sqrt{1-9 x}}$
1.45. $\int a^{5 x} d x$
1.46. $\int \frac{x^{2}-4}{x^{2}+4} d x$
1.47. $\int \frac{x+3}{3 x-4} d x$
1.48. $\int \frac{x^{4} d x}{x^{2}+4}$
1.49. $\int \frac{d x}{x^{2}+5 x+10}$
1.50. $\int \frac{d x}{\sqrt{1-(3 x+2)^{2}}}$
1.51. $\int \frac{d x}{x(x+2)}$
1.52. $\int \cos ^{2} x d x$
1.53. $\int \frac{d x}{\sin x}$
1.54. $\int \cos 2 x \cos 5 x d x$
1.55. $\int \sin ^{3} x d x$
1.56. $\int \cos ^{4} x d x$
1.57. $\int \cos ^{5} x d x$
1.58. $\int \operatorname{ctg}^{3} x d x$
1.59. $\int \cos x \sin 4 x \cos 6 x d x$
1.60. $\int \frac{d x}{\cos ^{4} x}$
1.61. $\int \frac{d x}{1-\cos x}$
1.62. $\int \frac{d x}{1+\sin x}$
1.63. $\int x \sin x d x$
1.64. $\int \operatorname{arctg} x d x$
1.65. $\int \arcsin x d x$
1.66. $\int x 2^{x} d x$
1.67. $\int x \operatorname{tg}^{2} x d x$
1.68. $\int x^{n} \ln x d x$
1.69. $\int \ln \left(x^{2}+4\right) d x$
1.70. $\int x^{2} e