Find an anti derivative (or integral) of the following function by the method of inspection: (ax + b)2
Solution and Explanation
The anti derivative of (ax + b)2 is the function of x whose derivative is (ax + b)2 .
It is known that,
\(\frac{d}{dx}\)(ax + b)3= 3a (ax + b)2
⇒ (ax + b)2 = \(\frac{1}{3a}\frac{d}{dx}\)(ax + b)3
∴ (ax + b)2 = \(\frac{d}{dx}\bigg(\frac{1}{3a}(ax+b)^3\bigg)\)
Therefore, the anti derivative of (ax + b)2 is \(\frac{1}{3a}\)(ax + b)3 .
Top Questions on integral
- Find the value of the integral: \[ \int_0^\pi \sin^2(x) \, dx. \]
Let \( f : (0, \infty) \to \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0 \), } \[ \int_0^a f(x) \, dx = f(a), \quad f(1) = 1, \quad f(16) = \frac{1}{8}, \quad \text{then } 16 - f^{-1}\left( \frac{1}{16} \right) \text{ is equal to:}\]
- Let $ f(x) $ be a positive function and $I_1 = \int_{-\frac{1}{2}}^1 2x \, f\left(2x(1-2x)\right) dx$ and $I_2 = \int_{-1}^2 f\left(x(1-x)\right) dx.$ Then the value of $\frac{I_2}{I_1}$ is equal to ____
- The value of the definite integral \( \int_0^{\pi} \sin^2 x \, dx \) is:
- Evaluate the integral: \[ \int \frac{\sin(2x)}{\sin(x)} \, dx \]
Questions Asked in CBSE CLASS XII exam
- Derive an expression for the torque acting on a rectangular current loop suspended in a uniform magnetic field.
- How do you convert:
Chlorobenzene to biphenyl- CBSE CLASS XII - 2025
- Organic Reactions
- A capacitor is charged by a battery to a potential difference \( V \). It is disconnected from the battery and connected across another identical uncharged capacitor. Calculate the ratio of total energy stored in the combination to the initial energy stored in the capacitor.
- CBSE CLASS XII - 2025
- Capacitors and Capacitance
- “There is a limit to the amount of charge that can be stored on a given capacitor.” Explain.
- CBSE CLASS XII - 2025
- Capacitors and Capacitance
- Mention the number of chromosomes at each stage. Correlate the life phases of the individual with the stages of the process.
- CBSE CLASS XII - 2025
- Pollination
Concepts Used:
Integral
The representation of the area of a region under a curve is called to be as integral. The actual value of an integral can be acquired (approximately) by drawing rectangles.
- The definite integral of a function can be shown as the area of the region bounded by its graph of the given function between two points in the line.
- The area of a region is found by splitting it into thin vertical rectangles and applying the lower and the upper limits, the area of the region is summarized.
- An integral of a function over an interval on which the integral is described.
Also, F(x) is known to be a Newton-Leibnitz integral or antiderivative or primitive of a function f(x) on an interval I.
F'(x) = f(x)
For every value of x = I.
Types of Integrals:
Integral calculus helps to resolve two major types of problems:
- The problem of getting a function if its derivative is given.
- The problem of getting the area bounded by the graph of a function under given situations.