1)∫
6x
2
+3
3x−5
dx=∫
16x
2
+5
3xdx
−∫
16x
2
+5
5dx
=
=
32
3
∫
16x
2
+5
32xdx
−5∫
(4x)
2
+(
5
)
2
dx
=
=
32
3
∫
(16x
2
+5)
2
1
d(16x
2
+5)
−
4
5
∫
(4x)
2
+(
5
)
2
d(4x)
=
=
32
3
2
1
(16x
2
+5)
2
1
−
4
5
×
5
1
arctg(
5
4x
)+C=
=
16
3
16x
2
+5
−
4
5
arctg(
5
4x
)+C
\begin{gathered}2)\int\limits( \sqrt[4]{3 + 8x} + \frac{3}{ {(11x + 1)}^{5} } )dx = \\ = \frac{1}{8} \int\limits {(3 + 8x)}^{ \frac{1}{4} } d(3x + 8) + \frac{3}{11} \int\limits \frac{d(11x + 1)}{ {(11x + 1)}^{5} } = \\ = \frac{1}{8} \times \frac{ {(3 + 8x)}^{ \frac{5}{4} } }{ \frac{5}{4} } + \frac{3}{11} \times \frac{ {(11x + 1)}^{ - 4} }{( - 4)} + C = \\ = \frac{1}{10} \sqrt[4]{ {(8x + 3)}^{5} } - \frac{3}{44 {(11x + 1)}^{4} } + C\end{gathered}
2)∫(
4
3+8x
+
(11x+1)
5
3
)dx=
=
8
1
∫(3+8x)
4
1
d(3x+8)+
11
3
∫
(11x+1)
5
d(11x+1)
=
=
8
1
×
4
5
(3+8x)
4
5
+
11
3
×
(−4)
(11x+1)
−4
+C=
=
10
1
4
(8x+3)
5
−
44(11x+1)
4
3
+C
1)∫
6x
2
+3
3x−5
dx=∫
16x
2
+5
3xdx
−∫
16x
2
+5
5dx
=
=
32
3
∫
16x
2
+5
32xdx
−5∫
(4x)
2
+(
5
)
2
dx
=
=
32
3
∫
(16x
2
+5)
2
1
d(16x
2
+5)
−
4
5
∫
(4x)
2
+(
5
)
2
d(4x)
=
=
32
3
2
1
(16x
2
+5)
2
1
−
4
5
×
5
1
arctg(
5
4x
)+C=
=
16
3
16x
2
+5
−
4
5
arctg(
5
4x
)+C
\begin{gathered}2)\int\limits( \sqrt[4]{3 + 8x} + \frac{3}{ {(11x + 1)}^{5} } )dx = \\ = \frac{1}{8} \int\limits {(3 + 8x)}^{ \frac{1}{4} } d(3x + 8) + \frac{3}{11} \int\limits \frac{d(11x + 1)}{ {(11x + 1)}^{5} } = \\ = \frac{1}{8} \times \frac{ {(3 + 8x)}^{ \frac{5}{4} } }{ \frac{5}{4} } + \frac{3}{11} \times \frac{ {(11x + 1)}^{ - 4} }{( - 4)} + C = \\ = \frac{1}{10} \sqrt[4]{ {(8x + 3)}^{5} } - \frac{3}{44 {(11x + 1)}^{4} } + C\end{gathered}
2)∫(
4
3+8x
+
(11x+1)
5
3
)dx=
=
8
1
∫(3+8x)
4
1
d(3x+8)+
11
3
∫
(11x+1)
5
d(11x+1)
=
=
8
1
×
4
5
(3+8x)
4
5
+
11
3
×
(−4)
(11x+1)
−4
+C=
=
10
1
4
(8x+3)
5
−
44(11x+1)
4
3
+C
