Question:
$\int \log_e (a^x + x^a) dx$ is equal to
$\int \log_e (a^x + x^a) dx$ is equal to
Show Hint
$\int a^x dx = \frac{a^x}{\ln a} + c$, $\int x^n dx = \frac{x^{n+1}}{n+1} + c$.
Updated On: May 2, 2026
- $\frac{x^{a+1}}{a+1} + c$
- $\frac{x^{a+1}}{a+1} + a^x \log a + c$
- $x^{a+1} + c$
- $\frac{x^{a+1}}{a+1} + a^x \frac{1}{\log a} + c$
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept:
The integral $\int \log(a^x + x^a) dx$ does not have a simple elementary form. The options suggest it's the integral of the sum, not the log. So probably the question is $\int (a^x + x^a) dx$.
Step 2: Detailed Explanation:
If the integrand is $a^x + x^a$, then $\int a^x dx = \frac{a^x}{\log a} + c$, and $\int x^a dx = \frac{x^{a+1}}{a+1} + c$. So the integral is $\frac{x^{a+1}}{a+1} + \frac{a^x}{\log a} + c$, which matches option (D).
Step 3: Final Answer:
The integral is $\frac{x^{a+1}}{a+1} + \frac{a^x}{\log a} + c$.
The integral $\int \log(a^x + x^a) dx$ does not have a simple elementary form. The options suggest it's the integral of the sum, not the log. So probably the question is $\int (a^x + x^a) dx$.
Step 2: Detailed Explanation:
If the integrand is $a^x + x^a$, then $\int a^x dx = \frac{a^x}{\log a} + c$, and $\int x^a dx = \frac{x^{a+1}}{a+1} + c$. So the integral is $\frac{x^{a+1}}{a+1} + \frac{a^x}{\log a} + c$, which matches option (D).
Step 3: Final Answer:
The integral is $\frac{x^{a+1}}{a+1} + \frac{a^x}{\log a} + c$.
Was this answer helpful?
0
0
Top MET Mathematics Questions
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- The equation $3^{3x + 4} = 9^{2x - 2},\ x>0$ has the solution
- A parallelogram is constructed on the vectors $\mathbf{a} = 3\alpha - \beta$, $\mathbf{b} = \alpha + 3\beta$. If $|\alpha| = |\beta| = 2$ and the angle between $\alpha$ and $\beta$ is $\dfrac{\pi}{3}$, then the length of a diagonal of the parallelogram is
- Every group of order 7 is
- Let $U_{n} = 2 + 2^{3} + 2^{5} + \cdots + 2^{2n+1}$ and $V_{n} = 1 + 4 + 4^{2} + \cdots + 4^{n-1}$. Then $\displaystyle\lim_{n \to \infty} \dfrac{U_n}{V_n}$ is equal to
View More Questions
Top MET Calculus Questions
- The minimum value of $x^{2} + \dfrac{1}{1 + x^{2}}$ is at}
- Derivative of $x^{x}$ is
- The value of $a$ for which $f(x) = a\sin x + \dfrac{1}{3}\sin 3x$ has an extremum at $x = \dfrac{\pi}{3}$ is
- If $|f(x)|$ is continuous at $x = a$, then $f(x)$
- The area in square units enclosed by the curve $x^{2}y = 36$, the $x$-axis and the lines $x = 6$ and $x = 9$ is
View More Questions
Top MET Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- Which of the following product is formed in the reaction $ CH_3MgBr {->[(i)CO_2][(ii)H_2O]} ? $
View More Questions