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Solve the trigonometric integral $\int\frac{\sec\left(x\right)}{3\tan\left(x\right)+7}dx$

Step-by-step Solution

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Final answer to the problem

$\frac{13}{99}\ln\left|0.6593962+\tan\left(\frac{x}{2}\right)\right|-\frac{13}{99}\ln\left|-1.516539+\tan\left(\frac{x}{2}\right)\right|+C_0$
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Step-by-step Solution

How should I solve this problem?

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  • Integrate by partial fractions
  • Integrate by substitution
  • Integrate by parts
  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using trigonometric identities
  • Integrate using basic integrals
  • Product of Binomials with Common Term
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1

Rewrite the trigonometric expression $\frac{\sec\left(x\right)}{3\tan\left(x\right)+7}$ inside the integral

$\int\frac{\sec\left(x\right)}{\frac{3\sin\left(x\right)+7\cos\left(x\right)}{\cos\left(x\right)}}dx$

Learn how to solve problems step by step online.

$\int\frac{\sec\left(x\right)}{\frac{3\sin\left(x\right)+7\cos\left(x\right)}{\cos\left(x\right)}}dx$

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Learn how to solve problems step by step online. Solve the trigonometric integral int(sec(x)/(3tan(x)+7))dx. Rewrite the trigonometric expression \frac{\sec\left(x\right)}{3\tan\left(x\right)+7} inside the integral. Simplify the expression inside the integral. We can solve the integral \int\frac{1}{3\sin\left(x\right)+7\cos\left(x\right)}dx by applying the method Weierstrass substitution (also known as tangent half-angle substitution) which converts an integral of trigonometric functions into a rational function of t by setting the substitution. Hence.

Final answer to the problem

$\frac{13}{99}\ln\left|0.6593962+\tan\left(\frac{x}{2}\right)\right|-\frac{13}{99}\ln\left|-1.516539+\tan\left(\frac{x}{2}\right)\right|+C_0$

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Function Plot

Plotting: $\frac{13}{99}\ln\left|0.6593962+\tan\left(\frac{x}{2}\right)\right|-\frac{13}{99}\ln\left|-1.516539+\tan\left(\frac{x}{2}\right)\right|+C_0$

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5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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