{30 x¹4}+15^{x} 225}{x¹5}+15^{x}} d x=

Correct Answer is : log⁡(x15+15x)2+c ≤ft(x¹5}+15^{x})²+clog(x15+15x)2+c, where c is the constant of integration

x¹5}+15 x¹4}+30 x¹3}+c^{}, where c is the constant of integration

{ ≤ft(x¹5}+15^{x})}{2}+c, where c is the constant of integration

≤ft(x¹5}+15^{x})²+c, where c is the constant of integration

≤ft(x¹5}+15^{x})+c, where c is the constant of integration

Correct Answer is : log⁡(x15+15x)2+c ≤ft(x¹5}+15^{x})²+clog(x15+15x)2+c, where c is the constant of integration

Test Index

MHTCET 13 Apr 2026 Shift 1 Paper

Question : 127 of 150

Marks: +1, -0

\int \frac{30 x^{14}+15^{x} \log 225}{x^{15}+15^{x}} d x=

$x^{15}+15 x^{14}+30 x^{13}+c^{\prime}$ , where c is the constant of integration
$\frac{\log \left(x^{15}+15^{x}\right)}{2}+c$ , where c is the constant of integration
$\log \left(x^{15}+15^{x}\right)^{2}+c$ , where c is the constant of integration
$\log \left(x^{15}+15^{x}\right)+c$ , where c is the constant of integration

Solution:

Concept:

Recognize that the numerator is a multiple of the derivative of the denominator and use

\int \frac{f'(x)}{f(x)} dx = \log|f(x)| + c

Explanation:

Let

f(x) = x^{15} + 15^x

Then

f'(x) = 15x^{14} + 15^x \log 15

Also,

\log 225 = \log(15^2) = 2\log 15

So the numerator becomes

30x^{14} + 15^x(2\log 15) = 2(15x^{14} + 15^x \log 15)

Thus numerator

= 2 f'(x)

The integral is

I = \int \frac{2 f'(x)}{f(x)} dx

Using the standard formula,

I = 2 \log|f(x)| + c

This can be written as

I = \log\left(x^{15}+15^x\right)^2 + c

Answer:

Option C:

\log\left(x^{15}+15^x\right)^2 + c

, where

c

is the constant of integration.

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