# What is #f(x) = int sqrt(x+3) dx# if #f(1)=7 #?

now

By signing up, you agree to our Terms of Service and Privacy Policy

To find ( f(x) = \int \sqrt{x + 3} , dx ) when ( f(1) = 7 ), integrate ( \sqrt{x + 3} ) from ( x = 1 ) to ( x = a ), where ( a ) is a variable. Then solve for ( a ) using the given condition ( f(1) = 7 ). After finding ( a ), integrate ( \sqrt{x + 3} ) from ( x = 1 ) to ( x = a ) to get ( f(x) ).

First, integrate ( \sqrt{x + 3} ) from ( x = 1 ) to ( x = a ):

[ f(a) = \int_1^a \sqrt{x + 3} , dx ]

[ = \left[ \frac{2}{3}(x + 3)^{3/2} \right]_1^a ]

[ = \frac{2}{3}(a + 3)^{3/2} - \frac{2}{3}(1 + 3)^{3/2} ]

Given that ( f(1) = 7 ), substitute ( f(1) = 7 ) into the equation:

[ 7 = \frac{2}{3}(1 + 3)^{3/2} - \frac{2}{3}(1 + 3)^{3/2} ]

[ 7 = \frac{2}{3}(a + 3)^{3/2} - \frac{2}{3}(4)^{3/2} ]

[ 7 = \frac{2}{3}(a + 3)^{3/2} - \frac{2}{3}(8) ]

[ 7 = \frac{2}{3}(a + 3)^{3/2} - \frac{16}{3} ]

[ \frac{16}{3} + 7 = \frac{2}{3}(a + 3)^{3/2} ]

[ \frac{37}{3} = \frac{2}{3}(a + 3)^{3/2} ]

[ \frac{37}{2} = (a + 3)^{3/2} ]

[ \left( \frac{37}{2} \right)^{2/3} = a + 3 ]

[ a = \left( \frac{37}{2} \right)^{2/3} - 3 ]

Finally, integrate ( \sqrt{x + 3} ) from ( x = 1 ) to ( x = a ) to find ( f(x) ):

[ f(x) = \int_1^a \sqrt{x + 3} , dx ]

[ = \left[ \frac{2}{3}(x + 3)^{3/2} \right]_1^a ]

[ = \frac{2}{3}(a + 3)^{3/2} - \frac{2}{3}(1 + 3)^{3/2} ]

[ = \frac{2}{3}\left( \left( \frac{37}{2} \right)^{2/3} - 3 + 3 \right)^{3/2} - \frac{2}{3}(4)^{3/2} ]

By signing up, you agree to our Terms of Service and Privacy Policy

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7