Enter a problem...
Calculus Examples
Step 1
Integrate by parts using the formula , where and .
Step 2
Step 2.1
Combine and .
Step 2.2
Combine and .
Step 2.3
Combine and .
Step 2.4
Combine and .
Step 3
Split the single integral into multiple integrals.
Step 4
Step 4.1
Combine and .
Step 4.2
Combine and .
Step 4.3
Combine and .
Step 4.4
Combine and .
Step 5
Since is constant with respect to , move out of the integral.
Step 6
Since is constant with respect to , move out of the integral.
Step 7
Step 7.1
Let . Find .
Step 7.1.1
Differentiate .
Step 7.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 7.1.3
Differentiate using the Power Rule which states that is where .
Step 7.1.4
Multiply by .
Step 7.2
Substitute the lower limit in for in .
Step 7.3
Multiply by .
Step 7.4
Substitute the upper limit in for in .
Step 7.5
Cancel the common factor of .
Step 7.5.1
Cancel the common factor.
Step 7.5.2
Rewrite the expression.
Step 7.6
The values found for and will be used to evaluate the definite integral.
Step 7.7
Rewrite the problem using , , and the new limits of integration.
Step 8
Combine and .
Step 9
Since is constant with respect to , move out of the integral.
Step 10
Step 10.1
Multiply by .
Step 10.2
Multiply by .
Step 11
The integral of with respect to is .
Step 12
Combine and .
Step 13
Since is constant with respect to , move out of the integral.
Step 14
Step 14.1
Let . Find .
Step 14.1.1
Differentiate .
Step 14.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 14.1.3
Differentiate using the Power Rule which states that is where .
Step 14.1.4
Multiply by .
Step 14.2
Substitute the lower limit in for in .
Step 14.3
Multiply by .
Step 14.4
Substitute the upper limit in for in .
Step 14.5
Cancel the common factor of .
Step 14.5.1
Cancel the common factor.
Step 14.5.2
Rewrite the expression.
Step 14.6
The values found for and will be used to evaluate the definite integral.
Step 14.7
Rewrite the problem using , , and the new limits of integration.
Step 15
Combine and .
Step 16
Since is constant with respect to , move out of the integral.
Step 17
Step 17.1
Multiply by .
Step 17.2
Multiply by .
Step 18
Factor out .
Step 19
Using the Pythagorean Identity, rewrite as .
Step 20
Step 20.1
Let . Find .
Step 20.1.1
Differentiate .
Step 20.1.2
The derivative of with respect to is .
Step 20.2
Substitute the lower limit in for in .
Step 20.3
The exact value of is .
Step 20.4
Substitute the upper limit in for in .
Step 20.5
Simplify.
Step 20.5.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
Step 20.5.2
The exact value of is .
Step 20.5.3
Multiply by .
Step 20.6
The values found for and will be used to evaluate the definite integral.
Step 20.7
Rewrite the problem using , , and the new limits of integration.
Step 21
Split the single integral into multiple integrals.
Step 22
Apply the constant rule.
Step 23
By the Power Rule, the integral of with respect to is .
Step 24
Step 24.1
Combine and .
Step 24.2
Combine and .
Step 24.3
Combine and .
Step 25
Since is constant with respect to , move out of the integral.
Step 26
By the Power Rule, the integral of with respect to is .
Step 27
Step 27.1
To write as a fraction with a common denominator, multiply by .
Step 27.2
Combine and .
Step 27.3
Combine the numerators over the common denominator.
Step 27.4
Combine and .
Step 27.5
Combine and .
Step 27.6
Combine and .
Step 27.7
Multiply by .
Step 27.8
Cancel the common factor of and .
Step 27.8.1
Factor out of .
Step 27.8.2
Cancel the common factors.
Step 27.8.2.1
Factor out of .
Step 27.8.2.2
Cancel the common factor.
Step 27.8.2.3
Rewrite the expression.
Step 27.8.2.4
Divide by .
Step 28
Since is constant with respect to , move out of the integral.
Step 29
Step 29.1
Let . Find .
Step 29.1.1
Differentiate .
Step 29.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 29.1.3
Differentiate using the Power Rule which states that is where .
Step 29.1.4
Multiply by .
Step 29.2
Substitute the lower limit in for in .
Step 29.3
Multiply by .
Step 29.4
Substitute the upper limit in for in .
Step 29.5
Cancel the common factor of .
Step 29.5.1
Factor out of .
Step 29.5.2
Cancel the common factor.
Step 29.5.3
Rewrite the expression.
Step 29.6
The values found for and will be used to evaluate the definite integral.
Step 29.7
Rewrite the problem using , , and the new limits of integration.
Step 30
Combine and .
Step 31
Since is constant with respect to , move out of the integral.
Step 32
Step 32.1
Multiply by .
Step 32.2
Multiply by .
Step 33
The integral of with respect to is .
Step 34
Combine and .
Step 35
Step 35.1
Evaluate at and at .
Step 35.2
Evaluate at and at .
Step 35.3
Evaluate at and at .
Step 35.4
Evaluate at and at .
Step 35.5
Evaluate at and at .
Step 35.6
Simplify.
Step 35.6.1
Combine and .
Step 35.6.2
Cancel the common factor of .
Step 35.6.2.1
Cancel the common factor.
Step 35.6.2.2
Divide by .
Step 35.6.3
Combine and .
Step 35.6.4
Cancel the common factor of .
Step 35.6.4.1
Cancel the common factor.
Step 35.6.4.2
Divide by .
Step 35.6.5
Combine and .
Step 35.6.6
Rewrite as a product.
Step 35.6.7
Multiply by .
Step 35.6.8
Multiply by .
Step 35.6.9
Combine and .
Step 35.6.10
Cancel the common factor of and .
Step 35.6.10.1
Factor out of .
Step 35.6.10.2
Cancel the common factors.
Step 35.6.10.2.1
Factor out of .
Step 35.6.10.2.2
Cancel the common factor.
Step 35.6.10.2.3
Rewrite the expression.
Step 35.6.10.2.4
Divide by .
Step 35.6.11
Multiply by .
Step 35.6.12
Multiply by .
Step 35.6.13
To write as a fraction with a common denominator, multiply by .
Step 35.6.14
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 35.6.14.1
Multiply by .
Step 35.6.14.2
Multiply by .
Step 35.6.15
Combine the numerators over the common denominator.
Step 35.6.16
Simplify the numerator.
Step 35.6.16.1
Evaluate .
Step 35.6.16.2
Multiply by .
Step 35.6.16.3
Multiply by .
Step 35.6.16.4
Evaluate .
Step 35.6.16.5
Raise to the power of .
Step 35.6.16.6
Add and .
Step 35.6.17
Cancel the common factor of and .
Step 35.6.17.1
Factor out of .
Step 35.6.17.2
Cancel the common factors.
Step 35.6.17.2.1
Factor out of .
Step 35.6.17.2.2
Cancel the common factor.
Step 35.6.17.2.3
Rewrite the expression.
Step 35.6.17.2.4
Divide by .
Step 35.6.18
Multiply by .
Step 35.6.19
Cancel the common factor of and .
Step 35.6.19.1
Factor out of .
Step 35.6.19.2
Cancel the common factors.
Step 35.6.19.2.1
Factor out of .
Step 35.6.19.2.2
Cancel the common factor.
Step 35.6.19.2.3
Rewrite the expression.
Step 35.6.19.2.4
Divide by .
Step 35.6.20
Add and .
Step 35.6.21
Multiply by .
Step 35.6.22
Add and .
Step 35.6.23
Multiply by .
Step 35.6.24
Add and .
Step 35.6.25
Multiply by .
Step 35.6.26
Raise to the power of .
Step 35.6.27
Move the negative in front of the fraction.
Step 35.6.28
Write as a fraction with a common denominator.
Step 35.6.29
Combine the numerators over the common denominator.
Step 35.6.30
Subtract from .
Step 35.6.31
Multiply by .
Step 35.6.32
One to any power is one.
Step 35.6.33
To write as a fraction with a common denominator, multiply by .
Step 35.6.34
Combine and .
Step 35.6.35
Combine the numerators over the common denominator.
Step 35.6.36
Simplify the numerator.
Step 35.6.36.1
Multiply by .
Step 35.6.36.2
Add and .
Step 35.6.37
Move the negative in front of the fraction.
Step 35.6.38
Multiply by .
Step 35.6.39
Multiply by .
Step 35.6.40
Combine the numerators over the common denominator.
Step 35.6.41
Add and .
Step 35.6.42
Raising to any positive power yields .
Step 35.6.43
Cancel the common factor of and .
Step 35.6.43.1
Factor out of .
Step 35.6.43.2
Cancel the common factors.
Step 35.6.43.2.1
Factor out of .
Step 35.6.43.2.2
Cancel the common factor.
Step 35.6.43.2.3
Rewrite the expression.
Step 35.6.43.2.4
Divide by .
Step 35.6.44
Multiply by .
Step 35.6.45
Add and .
Step 35.6.46
Cancel the common factor of and .
Step 35.6.46.1
Factor out of .
Step 35.6.46.2
Factor out of .
Step 35.6.46.3
Factor out of .
Step 35.6.46.4
Cancel the common factors.
Step 35.6.46.4.1
Factor out of .
Step 35.6.46.4.2
Cancel the common factor.
Step 35.6.46.4.3
Rewrite the expression.
Step 35.6.47
Multiply by .
Step 35.6.48
Combine.
Step 35.6.49
Apply the distributive property.
Step 35.6.50
Cancel the common factor of .
Step 35.6.50.1
Cancel the common factor.
Step 35.6.50.2
Rewrite the expression.
Step 35.6.51
Simplify the numerator.
Step 35.6.51.1
Apply the product rule to .
Step 35.6.51.2
Raise to the power of .
Step 35.6.52
Rewrite as a product.
Step 35.6.53
Multiply by .
Step 35.6.54
Multiply by .
Step 35.6.55
Combine and .
Step 35.6.56
Combine and .
Step 35.6.57
Multiply by .
Step 35.6.58
Multiply by .
Step 35.6.59
Multiply by .
Step 35.6.60
Combine.
Step 35.6.61
Apply the distributive property.
Step 35.6.62
Cancel the common factor of .
Step 35.6.62.1
Cancel the common factor.
Step 35.6.62.2
Rewrite the expression.
Step 35.6.63
Multiply by .
Step 35.6.64
Multiply by .
Step 35.6.65
To write as a fraction with a common denominator, multiply by .
Step 35.6.66
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 35.6.66.1
Multiply by .
Step 35.6.66.2
Multiply by .
Step 35.6.67
Combine the numerators over the common denominator.
Step 35.6.68
Multiply by .
Step 36
Step 36.1
The exact value of is .
Step 36.2
The exact value of is .
Step 37
Step 37.1
Simplify the numerator.
Step 37.1.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 37.1.2
The exact value of is .
Step 37.2
Divide by .
Step 37.3
Multiply by .
Step 37.4
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 37.5
The exact value of is .
Step 37.6
Multiply by .
Step 37.7
Simplify each term.
Step 37.7.1
Simplify the numerator.
Step 37.7.1.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 37.7.1.2
The exact value of is .
Step 37.7.1.3
Raising to any positive power yields .
Step 37.7.2
Divide by .
Step 37.7.3
Divide by .
Step 37.8
Add and .
Step 37.9
Add and .
Step 37.10
Add and .
Step 37.11
Multiply .
Step 37.11.1
Multiply by .
Step 37.11.2
Raise to the power of .
Step 37.11.3
Raise to the power of .
Step 37.11.4
Use the power rule to combine exponents.
Step 37.11.5
Add and .
Step 37.11.6
Multiply by .
Step 37.12
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
Step 37.13
The exact value of is .
Step 37.14
Multiply by .
Step 37.15
Multiply by .
Step 37.16
Add and .
Step 37.17
Multiply by .
Step 37.18
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 37.19
The exact value of is .
Step 37.20
Multiply by .
Step 37.21
Add and .
Step 37.22
Multiply by .
Step 37.23
Add and .
Step 37.24
Simplify each term.
Step 37.24.1
Cancel the common factor of and .
Step 37.24.1.1
Factor out of .
Step 37.24.1.2
Cancel the common factors.
Step 37.24.1.2.1
Factor out of .
Step 37.24.1.2.2
Cancel the common factor.
Step 37.24.1.2.3
Rewrite the expression.
Step 37.24.2
Move the negative in front of the fraction.
Step 37.25
To write as a fraction with a common denominator, multiply by .
Step 37.26
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 37.26.1
Multiply by .
Step 37.26.2
Multiply by .
Step 37.27
Combine the numerators over the common denominator.
Step 37.28
Add and .
Step 37.29
To write as a fraction with a common denominator, multiply by .
Step 37.30
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 37.30.1
Multiply by .
Step 37.30.2
Multiply by .
Step 37.31
Combine the numerators over the common denominator.
Step 37.32
Multiply by .
Step 38
Step 38.1
Apply the distributive property.
Step 38.2
Multiply by .
Step 38.3
Multiply by .
Step 38.4
Subtract from .
Step 39
The result can be shown in multiple forms.
Exact Form:
Decimal Form: