# What is #f(x) = int -cos^3x +tanx dx# if #f(pi)=-2 #?

This can be split into two integrals:

To find the first, do what follows:

Thus, the whole expression equals

Thus, we obtain

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

To find ( f(x) = \int -\cos^3(x) + \tan(x) , dx ) given that ( f(\pi) = -2 ), we first need to evaluate the integral and then solve for the constant of integration using the given condition.

Integrate ( -\cos^3(x) + \tan(x) ) with respect to ( x ):

[ \int -\cos^3(x) + \tan(x) , dx = -\int \cos^3(x) , dx + \int \tan(x) , dx ]

[ = -\int (1 - \sin^2(x))\cos(x) , dx + \int \tan(x) , dx ]

[ = -\int \cos(x) , dx + \int \sin^2(x)\cos(x) , dx + \int \tan(x) , dx ]

[ = -\sin(x) + \int \sin^2(x)\cos(x) , dx + \int \tan(x) , dx ]

Now, let ( u = \sin(x) ), so ( du = \cos(x) , dx ):

[ = -\sin(x) + \int u^2 , du + \int \tan(x) , dx ]

[ = -\sin(x) + \frac{u^3}{3} + \int \tan(x) , dx ]

[ = -\sin(x) + \frac{\sin^3(x)}{3} + \ln|\sec(x)| + C ]

Now, we can use the given condition ( f(\pi) = -2 ) to find the constant of integration ( C ):

[ f(\pi) = -\sin(\pi) + \frac{\sin^3(\pi)}{3} + \ln|\sec(\pi)| + C = -2 ]

[ 0 + 0 + \ln|(-1)| + C = -2 ]

[ \ln(1) + C = -2 ]

[ C = -2 ]

Therefore, the function ( f(x) = \int -\cos^3(x) + \tan(x) , dx ) with the given condition is:

[ f(x) = -\sin(x) + \frac{\sin^3(x)}{3} + \ln|\sec(x)| - 2 ]

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

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is #F(x) = int 3x^2-e^(2-x) dx# if #F(0) = 1 #?
- How do you find the antiderivative of #int 1/root3(1-5t) dt#?
- How do you use u-substitution, solve for all the real value(s) of 'x' for the equation #x^4-3x^2=10#?
- How do you integrate #int sqrtx ln 2x dx # using integration by parts?
- How do you integrate #int (3x+7)/(x^4-16)dx# using partial fractions?

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