Integration Worksheet Solution1





In this page integration worksheet solution1 we are going to see solution of some practice questions of the integration worksheet.

Question 1

Integrate the following with respect to x  3 x³ + 7 x² - 2 x + 1

Solution:

Formula :

∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c

                        =  ∫(3 x³ + 7 x² - 2 x + 1) dx

Now we are going to write separate integration.

                        = ∫(3 x³) dx + ∫(7 x²) dx -  ∫(2 x) dx + ∫1 dx

Take constant term out of integration.

                        = 3 ∫x³ dx + 7 ∫ x² dx - 2 ∫ x dx + ∫ 1 dx

                        = 3 (x⁴/4) + 7 (x³/3) - 2 (x²/2) + x + C

                        = 3 x⁴/4 + 7x³/3 - x² + x + C

Answer is 3 x⁴/4 + 7x³/3 - x² + x + C


Question 2

Integrate the following with respect to x 7 x² + 2 x + 3/x⁵

Solution:

Formula :

∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c

                        =  ∫ (7 x² + 2 x + 3)/x⁵ dx

Now we are going to separate the denominator x⁵ for above numerator terms.

                        = ∫ (7 x²/x⁵) dx + ∫(2 x/x⁵) dx -  ∫ (3/x⁵) dx

                        = ∫ (7/x³ ) dx + ∫(2/x⁴) dx -  ∫ (3/x⁵) dx

                        = ∫ (7x⁻³) dx + ∫(2x⁻⁴) dx -  ∫ (3x⁻⁵) dx

                        = (7x⁻²)/(-2) + (2x⁻³)/(-3) - (3x⁻⁴/(-4)

                        = -7/2 x² - 2/3x³ + 3/4x⁴ + C

Answer is -7/2 x² - 2/3x³ + 3/4x⁴ + C


Question 3

Integrate the following with respect to x    (x - 1) (x + 2)

Solution:

Formula :

∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c

                        =  ∫ (x - 1) (x + 2) dx

We can integrate the above function using product rule.

                        = ∫ ( x² + 2 x - x - 2 ) dx

                        = ∫ ( x² + x - 2 ) dx

                        = ∫ x² dx + ∫ x dx - ∫ 2 dx

                        = (x³/3) + (x²/2) - 2 ∫ dx

                        = (x³/3) + (x²/2) - 2 x + C

Answer is (x³/3) + (x²/2) - 2 x + C

integration worksheet solution1 integration worksheet solution1