Reduction Theorem

Reduction Theorem Formulas help us in reducing the tedious work of calculation, the lengthy integrations, and reduce our mathematical computational work. These formulas are used to compute the higher order integrations. In this article, we will discuss the formulas for various types of expressions.

Reduction Theorem/Reduction for Basic Exponential Expressions

$\int x^{n} \cdot e^{m x} \cdot d x=\dfrac{1}{m} \cdot x^{n} \cdot e^{m x}-\dfrac{n}{m} \int x^{n-1} \cdot e^{m x} \cdot d x$

Statement of Reduction Theorem/Reduction for Logarithmic Expressions

• $\int \log ^{n} x \cdot d x=x \log ^{n} x-n \int \log ^{n-1} x \cdot d x$

• $\int x^{n} \log ^{m} x \cdot d x=\dfrac{x^{n+1} \log ^{m} x}{n+1}-\dfrac{m}{n+1} \int x^{n} \log ^{m-1} x \cdot d x$

Reduction Theorem/Reduction for Trigonometric Functions

• $\int \sin ^{n} x \cdot d x=-\dfrac{1}{n} \sin ^{n-1} x \cdot \cos x+\dfrac{n-1}{n} \int \sin ^{n-2} x \cdot d x$

• $\int \cos ^{n} x \cdot d x=\dfrac{1}{n} \cos ^{n-1} x \cdot \sin x+\dfrac{n-1}{n} \int \cos ^{n-2} x \cdot d x$

• $\int \sin ^{n} x \cdot \cos ^{m} x \cdot d x=\dfrac{\sin ^{n+1} x \cdot \cos ^{m-1} x}{n+m}+\dfrac{m-1}{n+m} \int \sin ^{n} x \cdot \cos ^{m-2} x \cdot d x$

• $\int \tan ^{n} x \cdot d x=\dfrac{1}{n-1} \cdot \tan ^{n-1} x-\int \tan ^{n-2} x \cdot d x$
  
  

Statement of Reduction Theorem/Reduction for Algebraic Expressions

$\int \dfrac{x^{n}}{a x^{n}+b} \cdot d x=\dfrac{x}{a}-\dfrac{b}{a} \int \dfrac{1}{a x^{n}+b} \cdot d x$

Sin function

Properties of Reduction

• Reduction Formulas are used to evaluate higher order integrals that cannot be evaluated otherwise.

• Reductions Formulas are applicable across a wide range of variables such as exponential, logarithmic, algebraic, trigonometric, etc.

Area Under the Curve

Solved Examples

1. Find the integral of \[\sin ^{5 x}\].
Ans:
Apply reduction formula:

$\Rightarrow-\dfrac{1}{5} \cdot \sin ^{4} x \cdot \cos x+\dfrac{4}{5} \int \sin ^{3} x \cdot d x \\$

$\Rightarrow-\dfrac{1}{5} \cdot \sin ^{4} x \cdot \cos x+\dfrac{4}{5}\left(\int \dfrac{(3 \sin x-\sin 3 x)}{4} \cdot d x\right) \\$

$\Rightarrow-\dfrac{1}{5} \cdot \sin ^{4} x \cdot \cos x+\dfrac{1}{5}\left(3 \int \sin x \cdot d x-\int \sin 3 x \cdot d x\right) \\$

$\Rightarrow-\dfrac{1}{5} \sin ^{4} x \cdot \cos x+\dfrac{1}{5}\left(-3 \cos x+\dfrac{\cos 3 x}{3}\right) \\$

$\Rightarrow-\dfrac{1}{5} \cdot \sin ^{4} x \cdot \cos x-\dfrac{3 \cos x}{5}+\dfrac{\cos 3 x}{15}$

Therefore, we have

$\Rightarrow \int \sin ^{5} x \cdot d x=-\dfrac{1}{5} \cdot \sin ^{4} x \cdot \cos x-\dfrac{3 \cos x}{5}+\dfrac{\cos 3 x}{15}$

2. Evaluate the integral of $x^{3} \log ^{2} x$.
Ans:
Applying the reduction formula:

$\int x^{n} \log ^{m} x \cdot d x=\dfrac{x^{n+1} \log ^{m} x}{n+1}-\dfrac{m}{n+1} \int x^{n} \log ^{m-1} x . \\$

$\int x^{3} \log ^{2} x \cdot d x=\dfrac{x^{4} \log ^{2} x}{4}-\dfrac{2}{4} \int x^{3} \log x \cdot d x \\$

$\Rightarrow \dfrac{x^{4} \log ^{2} x}{4}-\dfrac{1}{2}\left(\dfrac{x^{4} \log x}{4}-\dfrac{1}{4} \cdot \int x^{3} \cdot d x\right) \\$

$\Rightarrow \dfrac{x^{4} \log ^{2} x}{4}-\dfrac{x^{4} \log x}{8}+\dfrac{x^{4}}{32}+C$

Therefore, we have

$\Rightarrow \int x^{3} \log ^{2} x \cdot d x=\dfrac{x^{4} \log ^{2} x}{4}-\dfrac{x^{4} \log x}{8}+\dfrac{x^{4}}{32}+C$

3. Evaluate $\int_{0}^{2 a} x^{2} \sqrt{2 a x-x^{2}} d x$.
Ans:

So, $d x=-4 a \cos \theta \sin \theta d \theta$.

At $x=0,2 \operatorname{acos}^{2} \theta=0$ and so $\theta=\dfrac{\pi}{2}$.

When $x=2 a, 2 a \cos ^{2} \theta=2 a$ and so $\theta=0$.

Hence, we have

$\Rightarrow I=\int_{0}^{2 a} x^{2} \sqrt{2 a x-x^{2}} d x$

$\Rightarrow \int_{\dfrac{\pi}{2}}^{0} 4 a^{2} \cos ^{2} \theta \sqrt{4 a^{2} \cos ^{2} \theta-4 a^{2} \cos ^{4} \theta}(-4 a \cos \theta \sin \theta) d \theta$

$\Rightarrow \int_{0}^{\dfrac{\pi}{2}} 4 a^{2} \cos ^{2} \theta 2 a \cos \theta \sin \theta(4 a \cos \theta \sin \theta) d \theta$

$\Rightarrow 32 a^{4} \int_{0}^{\dfrac{\pi}{2}} \cos ^{4} \theta \sin ^{2} \theta d \theta$

$\Rightarrow 32 a^{4} \times \dfrac{1}{6} \times \dfrac{3}{4} \times \dfrac{1}{2} \times \dfrac{\pi}{2}$

$\Rightarrow \pi a^{4}$

Conclusion

In the article, we have discussed the detailed proof of Reduction Formulas and their applications. They are very helpful and reduce our computational work. So, reduction formulas are very important and can be derived using simple integration formulas; however, the proof is not important for us, so we need to remember all the formulas used to compute higher-order integrals.

Important Points to Remember

All the Reduction Formulas are important and are to be remembered.