Calculus Examples

Evaluate the Integral integral of xarctan(7x) with respect to x
Step 1
Integrate by parts using the formula , where and .
Step 2
Simplify.
Tap for more steps...
Step 2.1
Combine and .
Step 2.2
Combine and .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Simplify.
Tap for more steps...
Step 4.1
Combine and .
Step 4.2
Move to the left of .
Step 5
Since is constant with respect to , move out of the integral.
Step 6
Simplify.
Tap for more steps...
Step 6.1
Multiply by .
Step 6.2
Combine and .
Step 6.3
Move the negative in front of the fraction.
Step 7
Divide by .
Tap for more steps...
Step 7.1
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
++++
Step 7.2
Divide the highest order term in the dividend by the highest order term in divisor .
++++
Step 7.3
Multiply the new quotient term by the divisor.
++++
+++
Step 7.4
The expression needs to be subtracted from the dividend, so change all the signs in
++++
---
Step 7.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
++++
---
-
Step 7.6
The final answer is the quotient plus the remainder over the divisor.
Step 8
Split the single integral into multiple integrals.
Step 9
Apply the constant rule.
Step 10
Since is constant with respect to , move out of the integral.
Step 11
Since is constant with respect to , move out of the integral.
Step 12
Reorder and .
Step 13
Factor out of .
Tap for more steps...
Step 13.1
Factor out of .
Step 13.2
Factor out of .
Step 13.3
Factor out of .
Step 14
Since is constant with respect to , move out of the integral.
Step 15
Simplify.
Tap for more steps...
Step 15.1
Multiply by .
Step 15.2
Multiply by .
Step 16
Rewrite as .
Step 17
The integral of with respect to is .
Step 18
Simplify.
Tap for more steps...
Step 18.1
Simplify.
Tap for more steps...
Step 18.1.1
Multiply by the reciprocal of the fraction to divide by .
Step 18.1.2
Multiply by .
Step 18.1.3
Multiply by the reciprocal of the fraction to divide by .
Step 18.1.4
Move to the left of .
Step 18.2
Simplify.
Step 18.3
Simplify.
Tap for more steps...
Step 18.3.1
To write as a fraction with a common denominator, multiply by .
Step 18.3.2
Combine and .
Step 18.3.3
Combine the numerators over the common denominator.
Step 18.3.4
Multiply by .
Step 18.3.5
Combine and .
Step 18.3.6
Multiply by .
Step 18.3.7
Cancel the common factor of and .
Tap for more steps...
Step 18.3.7.1
Factor out of .
Step 18.3.7.2
Cancel the common factors.
Tap for more steps...
Step 18.3.7.2.1
Factor out of .
Step 18.3.7.2.2
Cancel the common factor.
Step 18.3.7.2.3
Rewrite the expression.
Step 18.3.7.2.4
Divide by .
Step 18.4
Simplify.
Tap for more steps...
Step 18.4.1
Apply the distributive property.
Step 18.4.2
Cancel the common factor of .
Tap for more steps...
Step 18.4.2.1
Factor out of .
Step 18.4.2.2
Factor out of .
Step 18.4.2.3
Cancel the common factor.
Step 18.4.2.4
Rewrite the expression.
Step 18.4.3
Cancel the common factor of .
Tap for more steps...
Step 18.4.3.1
Move the leading negative in into the numerator.
Step 18.4.3.2
Factor out of .
Step 18.4.3.3
Factor out of .
Step 18.4.3.4
Cancel the common factor.
Step 18.4.3.5
Rewrite the expression.
Step 18.4.4
Simplify each term.
Tap for more steps...
Step 18.4.4.1
Move the negative in front of the fraction.
Step 18.4.4.2
Multiply .
Tap for more steps...
Step 18.4.4.2.1
Multiply by .
Step 18.4.4.2.2
Multiply by .
Step 19
Reorder terms.