Step by step indefinite integral sample

Step by step solution of indefinite integral, returned by our online calculator is given below:

Indefinite integrals calculator
Find indefinite integral:
\(\displaystyle\int{\dfrac{4}{5\cdot x^2+7\cdot x+8}}\,\mathrm{d}x\)


Input interpretation
Find indefinite integral:
\(\displaystyle\int{\dfrac{4}{5\cdot x^2+7\cdot x+8}}\,\mathrm{d}x\)
Answer
\(\frac{8}{\sqrt{111}}\cdot \arctg\left(\frac{10\cdot x+7}{\sqrt{111}}\right)+Const\)

Step by step solution

Step 1
Integrate expression:
\(\displaystyle\int{\dfrac{4}{5\cdot x^2+7\cdot x+8}}\,\mathrm{d}x\)
Step 2
Factor out multiplier in denominator:
\(\displaystyle\int{\dfrac{4}{5\cdot \left(x^2+\frac{7\cdot x}{5}+\frac{8}{5}\right)}}\,\mathrm{d}x\)
Step 3
Factor out constants:
\(\dfrac{4}{5}\cdot\displaystyle\int{\dfrac{1}{x^2+\frac{7\cdot x}{5}+\frac{8}{5}}}\,\mathrm{d}x\)
Step 4
Complete the square in the denominator:
\(\dfrac{4}{5}\cdot\displaystyle\int{\dfrac{1}{\left(x+\frac{7}{10}\right)^2+\frac{111}{100}}}\,\mathrm{d}x\)
Step 5
Perform variable substitution:
\(u=x+\frac{7}{10}\) , then \(\mathrm{d}u=\mathrm{d}x\) and, therefore, \(\mathrm{d}x=\mathrm{d}u\):
\(\dfrac{4}{5}\cdot\displaystyle\int{\dfrac{1}{u^2+\frac{111}{100}}}\,\mathrm{d}u\)
Step 6
Using integrals identities \(\left(\int\frac{1}{u^2+a^2}=\frac{1}{a}\cdot \arctg\left(\frac{u}{a}\right)\right)\) , we get:
\(\dfrac{4}{5}\cdot\dfrac{1}{\sqrt{\dfrac{111}{100}}}\cdot\arctg\left(\frac{u}{\sqrt{\frac{111}{100}}}\right)\)
Step 7
Simplify:
\(\frac{8}{\sqrt{111}}\cdot\arctg\left(\frac{10\cdot u}{\sqrt{111}}\right)\)
Step 8
Substitute variable back:
\(\frac{8}{\sqrt{111}}\cdot\arctg\left(\frac{10\cdot x + 7}{\sqrt{111}}\right)\)
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See also:

Basic features of indefinite integral
Integration by substitution method

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