From Wikibooks, open books for an open world
Evaluate the following:
Solutions
Find the general antiderivative of the following:
Solutions
Find the anti-derivative or compute the integral depending on whether the integral is indefinite or definite.
14.
∫
0
π
/
4
tan
(
x
)
d
x
{\displaystyle \int _{0}^{\pi /4}\tan(x)\,dx}
.
ln
(
2
)
2
{\displaystyle {\frac {\ln(2)}{2}}}
ln
(
2
)
2
{\displaystyle {\frac {\ln(2)}{2}}}
15.
∫
1
/
2
1
e
2
x
−
1
2
x
−
1
d
x
{\displaystyle \int _{1/2}^{1}{\frac {e^{\sqrt {2x-1}}}{\sqrt {2x-1}}}\,dx}
.
e
−
1
{\displaystyle e-1}
e
−
1
{\displaystyle e-1}
16.
∫
−
3
−
6
8
x
x
2
−
5
d
x
{\displaystyle \int _{-3}^{-{\sqrt {6}}}{\frac {8x}{\sqrt {x^{2}-5}}}\,dx}
.
−
8
{\displaystyle -8}
−
8
{\displaystyle -8}
17.
∫
−
3
2
2
e
3
x
−
2
d
x
{\displaystyle \int -{\frac {3}{2}}{\sqrt {\frac {2}{e^{3x-2}}}}\,dx}
.
2
e
3
x
−
2
+
C
{\displaystyle {\sqrt {\frac {2}{e^{3x-2}}}}+C}
2
e
3
x
−
2
+
C
{\displaystyle {\sqrt {\frac {2}{e^{3x-2}}}}+C}
18.
∫
x
sec
(
x
2
−
5
)
tan
(
x
2
−
5
)
50
x
2
−
5
d
x
{\displaystyle \int {\frac {x\sec \left({\sqrt {x^{2}-5}}\right)\tan \left({\sqrt {x^{2}-5}}\right)}{50{\sqrt {x^{2}-5}}}}\,dx}
.
sec
(
x
2
−
5
)
50
+
C
{\displaystyle {\frac {\sec \left({\sqrt {x^{2}-5}}\right)}{50}}+C}
sec
(
x
2
−
5
)
50
+
C
{\displaystyle {\frac {\sec \left({\sqrt {x^{2}-5}}\right)}{50}}+C}
19.
∫
2
sec
2
(
ln
(
x
)
)
tan
(
ln
(
x
)
)
x
d
x
{\displaystyle \int {\frac {2\sec ^{2}\left(\ln(x)\right)\tan \left(\ln(x)\right)}{x}}\,dx}
.
tan
2
(
ln
(
x
)
)
+
C
{\displaystyle \tan ^{2}\left(\ln(x)\right)+C}
tan
2
(
ln
(
x
)
)
+
C
{\displaystyle \tan ^{2}\left(\ln(x)\right)+C}
20.
∫
(
e
x
−
1
)
x
e
x
−
2
+
x
e
x
−
1
e
x
ln
(
x
)
d
x
{\displaystyle \int \left(e^{x}-1\right)x^{e^{x}-2}+x^{e^{x}-1}e^{x}\ln(x)\,dx}
.
x
e
x
−
1
+
C
{\displaystyle x^{e^{x}-1}+C}
x
e
x
−
1
+
C
{\displaystyle x^{e^{x}-1}+C}
21.
∫
2
sec
2
(
ln
(
x
2
+
2
x
)
)
x
3
−
1
x
4
+
2
x
d
x
{\displaystyle \int 2\sec ^{2}\left(\ln \left(x^{2}+{\frac {2}{x}}\right)\right){\frac {x^{3}-1}{x^{4}+2x}}\,dx}
.
tan
(
ln
(
x
2
+
2
x
)
)
+
C
{\displaystyle \tan \left(\ln \left(x^{2}+{\frac {2}{x}}\right)\right)+C}
tan
(
ln
(
x
2
+
2
x
)
)
+
C
{\displaystyle \tan \left(\ln \left(x^{2}+{\frac {2}{x}}\right)\right)+C}
22.
∫
2
(
sin
(
x
)
−
cos
(
x
)
)
tan
(
sin
(
x
)
+
cos
(
x
)
)
sin
(
x
)
+
cos
(
x
)
d
x
{\displaystyle \int 2{\frac {\left(\sin(x)-\cos(x)\right)\tan \left({\sqrt {\sin(x)+\cos(x)}}\right)}{\sqrt {\sin(x)+\cos(x)}}}\,dx}
.
4
ln
|
cos
(
sin
(
x
)
+
cos
(
x
)
)
|
+
C
{\displaystyle 4\ln \left\vert \cos \left({\sqrt {\sin(x)+\cos(x)}}\right)\right\vert +C}
4
ln
|
cos
(
sin
(
x
)
+
cos
(
x
)
)
|
+
C
{\displaystyle 4\ln \left\vert \cos \left({\sqrt {\sin(x)+\cos(x)}}\right)\right\vert +C}
23.
∫
x
x
+
1
d
x
{\displaystyle \int {\frac {x}{\sqrt {x+1}}}\,dx}
.
2
3
(
x
−
2
)
x
+
1
+
C
{\displaystyle {\frac {2}{3}}(x-2){\sqrt {x+1}}+C}
2
3
(
x
−
2
)
x
+
1
+
C
{\displaystyle {\frac {2}{3}}(x-2){\sqrt {x+1}}+C}
24.
∫
x
+
1
x
(
x
+
ln
x
)
2
d
x
{\displaystyle \int {\frac {x+1}{x\left(x+\ln x\right)^{2}}}\,dx}
.
−
1
x
+
ln
x
+
C
{\displaystyle -{\frac {1}{x+\ln x}}+C}
−
1
x
+
ln
x
+
C
{\displaystyle -{\frac {1}{x+\ln x}}+C}
Solutions
Solutions
Solutions
