This calculator efficiently determines the definite integral of functions that follow the form $\int_{a}^{b} (Ax + B)^N dx$, a classic application of the $u$-substitution method. Quickly find the area under the curve between two defined bounds.
U Substitution Calculator
U Substitution Calculator Formula
Variables Explained
- Coefficient A: The multiplier of $x$ inside the inner function $u = Ax + B$.
- Constant B: The additive constant inside the inner function $u = Ax + B$.
- Exponent N: The power to which the inner function $(Ax + B)$ is raised. Note: $N$ cannot equal -1.
- Lower Bound (a): The starting point for the definite integral, $x=a$.
- Upper Bound (b): The ending point for the definite integral, $x=b$.
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What is U-Substitution?
U-Substitution, also known as Integration by Substitution or the reverse chain rule, is a fundamental technique in calculus used to find integrals. It simplifies complex integrals into forms that are easier to solve by replacing a part of the integrand (the function being integrated) with a new variable, $u$.
The core idea is to identify an inner function whose derivative is also present in the integral. By setting the inner function equal to $u$ and finding the derivative $du$, the original integral is transformed into a simpler integral involving $u$ and $du$. This allows for straightforward integration using standard power rules or trigonometric identities.
This method is essential for solving composite functions, making it one of the most powerful tools in an applied mathematician’s toolkit for solving real-world problems involving rates of change and accumulation.
How to Calculate U-Substitution (Example)
Let’s calculate the definite integral $\int_{0}^{1} (2x + 1)^3 dx$ (using $A=2, B=1, N=3, a=0, b=1$):
- Identify U and DU: Let $u = 2x + 1$. Then the derivative with respect to $x$ is $\frac{du}{dx} = 2$. Therefore, $du = 2dx$, or $dx = \frac{1}{2}du$.
- Find the Antiderivative: Substitute $u$ and $dx$ into the integral: $\int u^3 \left( \frac{1}{2} du \right) = \frac{1}{2} \int u^3 du$. Applying the power rule: $\frac{1}{2} \left( \frac{u^4}{4} \right) = \frac{u^4}{8}$.
- Substitute Back X: Replace $u$ with $(2x + 1)$: The antiderivative is $\frac{(2x + 1)^4}{8}$.
- Apply the Bounds: Evaluate the antiderivative at the upper bound ($x=1$) and the lower bound ($x=0$): $$\frac{(2(1) + 1)^4}{8} – \frac{(2(0) + 1)^4}{8} = \frac{3^4}{8} – \frac{1^4}{8}$$
- Final Result: $\frac{81}{8} – \frac{1}{8} = \frac{80}{8} = 10$.
Frequently Asked Questions (FAQ)
If $N = -1$, the power rule for integration fails. The integral becomes $\int (Ax + B)^{-1} dx = \int \frac{A}{Ax+B} \frac{1}{A} dx$. Using $u=Ax+B$, this results in $\frac{1}{A} \int \frac{1}{u} du = \frac{1}{A} \ln|u| + C$. Our calculator handles this case as an error for simplicity.
Is u-substitution the same as the reverse chain rule?Yes, $u$-substitution is essentially the formal procedure for reversing the chain rule in differentiation. The chain rule is used to differentiate composite functions, and $u$-substitution is used to integrate them.
When should I use u-substitution?You should use $u$-substitution when the integrand is a composite function, specifically when you see a function $f(x)$ multiplied by its derivative $f'(x)$ (or a constant multiple of it).
Does this method work for definite integrals?Yes. For definite integrals, you have two options: either substitute $u$ back with $x$ before evaluating the bounds, or change the integration bounds from $x$-values to $u$-values when you make the substitution.