Advertisements
Advertisements
Question
If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).
Advertisements
Solution
f'(x) = x2 + 5 ...(Given)
∴ f(x) = ∫f'(x) dx
∴ f(x) = ∫(x2 + 5) dx
∴ f(x) = ∫ x2 dx + 5 ∫ dx
∴ f(x) = `"x"^3/3 + 5"x" + "c"` ....(i)
Substitute x = 0, f(0) = −1 ...(Given)
∴ f(x) = `"x"^3/3 + 5"x" + "c"`
∴ f(0) = `0^3/3 + 5(0) + "c"`
∴ −1 = 0 + 0 + c
∴ c = −1
Substituting c = – 1 in (i), we get,
∴ f(x) = `"x"^3/3 + 5"x" + (− 1)`
∴ f(x) = `"x"^3/3 + 5"x" − 1`
APPEARS IN
RELATED QUESTIONS
Evaluate :
`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`
Integrate the functions:
`e^(2x+3)`
Write a value of\[\int\frac{\sec^2 x}{\left( 5 + \tan x \right)^4} dx\]
Write a value of \[\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }\]
Evaluate: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
The value of \[\int\frac{1}{x + x \log x} dx\] is
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`
Evaluate the following integrals : `int sin x/cos^2x dx`
Evaluate the following integrals: `int sin 4x cos 3x dx`
Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`
Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`
Evaluate the following : `int (1)/(4 + 3cos^2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
Evaluate `int (3"x"^3 - 2sqrt"x")/"x"` dx
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 + 8))` dx
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
Evaluate `int 1/((2"x" + 3))` dx
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
State whether the following statement is True or False:
`int"e"^(4x - 7) "d"x = ("e"^(4x - 7))/(-7) + "c"`
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
`int (x + sinx)/(1 + cosx)dx` is equal to ______.
Evaluate:
`int sin^2(x/2)dx`
