Math virtual assistant

About Snapxam Calculators Topics Go Premium
ENGESP

Integration by trigonometric substitution Calculator

Get detailed step by step solutions to your math problems with our online calculator. Sharpen your math skills and learn step by step with our math solver. Check out more calculators here.

Go!
1
2
3
4
5
6
7
8
9
0
x
y
(◻)
◻/◻
2

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

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

1

Example

$\int\sqrt{x^2+4}dx$
2

Solve the integral $\int\sqrt{4+x^2}dx$ by trigonometric substitution using the substitution

$\begin{matrix}x=2\tan\left(\theta\right) \\ dx=2\sec\left(\theta\right)^2d\theta\end{matrix}$
3

Substituting in the original integral, we get

$\int2\sqrt{4+4\tan\left(\theta\right)^2}\sec\left(\theta\right)^2d\theta$
4

Take the constant out of the integral

$2\int\sqrt{4+4\tan\left(\theta\right)^2}\sec\left(\theta\right)^2d\theta$
5

Factor by the greatest common divisor $4$

$2\int\sqrt{4\left(1+1\tan\left(\theta\right)^2\right)}\sec\left(\theta\right)^2d\theta$
6

Any expression multiplied by $1$ is equal to itself

$2\int\sqrt{4\left(1+\tan\left(\theta\right)^2\right)}\sec\left(\theta\right)^2d\theta$
7

The power of a product is equal to the product of it's factors raised to the same power

$2\int2\sec\left(\theta\right)^2\sqrt{1+\tan\left(\theta\right)^2}d\theta$
8

Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$

$2\int2\sec\left(\theta\right)^2\sec\left(\theta\right)d\theta$
9

When multiplying exponents with same base you can add the exponents

$2\int2\sec\left(\theta\right)^{3}d\theta$
10

Take the constant out of the integral

$4\int\sec\left(\theta\right)^{3}d\theta$
11

Simplify the integral of secant applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$

$4\left(\frac{1}{2}\int\sec\left(\theta\right)d\theta+\frac{\sec\left(\theta\right)^{2}\sin\left(\theta\right)}{2}\right)$
12

Expressing the result of the integral in terms of the original variable

$4\left(\frac{1}{2}\int\sec\left(\theta\right)d\theta+\frac{\frac{x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}}{2}\right)$
13

Simplifying the fraction

$4\left(\frac{1}{2}\int\sec\left(\theta\right)d\theta+\frac{x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{2\sqrt{4+x^2}}\right)$
14

Taking out the constant $2$ from the fraction's denominator

$4\left(\frac{1}{2}\int\sec\left(\theta\right)d\theta+\frac{1}{2}\cdot\frac{x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}\right)$
15

The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$

$4\left(\frac{1}{2}\ln\left(\tan\left(\theta\right)+\sec\left(\theta\right)\right)+\frac{1}{2}\cdot\frac{x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}\right)$
16

Expressing the result of the integral in terms of the original variable

$4\left(\frac{1}{2}\ln\left(\frac{x}{2}+\frac{\sqrt{4+x^2}}{2}\right)+\frac{\frac{1}{2}x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}\right)$
17

Add fraction's numerators with common denominators: $\frac{\sqrt{4+x^2}}{2}$ and $\frac{x}{2}$

$4\left(\frac{1}{2}\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+\frac{\frac{1}{2}x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}\right)$
18

Multiply $\left(\frac{\frac{1}{2}x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}+\frac{1}{2}\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)\right)$ by $4$

$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+4\frac{\frac{1}{2}x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}$
19

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+4\frac{\frac{1}{2}x\left(\frac{4+x^2}{4}\right)}{\sqrt{4+x^2}}$
20

Simplify the fraction

$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+4\frac{\frac{1}{8}x\left(4+x^2\right)}{\sqrt{4+x^2}}$
21

Simplifying the fraction by $4+x^2$

$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+4\frac{\frac{1}{8}x}{\frac{1}{\sqrt{4+x^2}}}$
22

Simplifying the fraction

$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+4\cdot \frac{1}{8}\sqrt{4+x^2}x$
23

Multiply $\frac{1}{8}$ times $4$

$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+\frac{1}{2}\sqrt{4+x^2}x$
24

The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator

$2\left(\ln\left(2\right)\left(-1\right)+\ln\left(x+\sqrt{4+x^2}\right)\right)+\frac{1}{2}\sqrt{4+x^2}x$
25

Calculating the natural logarithm of $2$

$2\left(\frac{4}{\sqrt{3}}\left(-1\right)+\ln\left(x+\sqrt{4+x^2}\right)\right)+\frac{1}{2}\sqrt{4+x^2}x$
26

Multiply $-1$ times $\frac{4}{\sqrt{3}}$

$2\left(\ln\left(x+\sqrt{4+x^2}\right)-\frac{4}{\sqrt{3}}\right)+\frac{1}{2}\sqrt{4+x^2}x$
27

Multiply $\left(\ln\left(x+\sqrt{4+x^2}\right)+-\frac{4}{\sqrt{3}}\right)$ by $2$

$\frac{1}{2}\sqrt{4+x^2}x-\frac{17}{\sqrt{3}}+2\ln\left(x+\sqrt{4+x^2}\right)$
28

Add the constant of integration

$\frac{1}{2}\sqrt{4+x^2}x-\frac{17}{\sqrt{3}}+2\ln\left(x+\sqrt{4+x^2}\right)+C_0$

Struggling with math?

Access detailed step by step solutions to millions of problems, growing every day!